statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
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convex.integral_mem [is_probability_measure μ] (hs : convex ℝ s) (hsc : is_closed s)
(hf : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) :
∫ x, f x ∂μ ∈ s | begin
borelize E,
rcases hfi.ae_strongly_measurable with ⟨g, hgm, hfg⟩,
haveI : separable_space (range g ∩ s : set E) :=
(hgm.is_separable_range.mono (inter_subset_left _ _)).separable_space,
obtain ⟨y₀, h₀⟩ : (range g ∩ s).nonempty,
{ rcases (hf.and hfg).exists with ⟨x₀, h₀⟩,
exact ⟨f x₀, by simp onl... | lemma | convex.integral_mem | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"closure",
"convex",
"ennreal.one_to_real",
"ennreal.to_real_nonneg",
"ennreal.to_real_sum",
"is_closed",
"subset_closure"
] | If `μ` is a probability measure on `α`, `s` is a convex closed set in `E`, and `f` is an
integrable function sending `μ`-a.e. points to `s`, then the expected value of `f` belongs to `s`:
`∫ x, f x ∂μ ∈ s`. See also `convex.sum_mem` for a finite sum version of this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.average_mem [is_finite_measure μ] (hs : convex ℝ s) (hsc : is_closed s) (hμ : μ ≠ 0)
(hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) :
⨍ x, f x ∂μ ∈ s | begin
haveI : is_probability_measure ((μ univ)⁻¹ • μ),
from is_probability_measure_smul hμ,
refine hs.integral_mem hsc (ae_mono' _ hfs) hfi.to_average,
exact absolutely_continuous.smul (refl _) _
end | lemma | convex.average_mem | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"convex",
"is_closed"
] | If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
`⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.set_average_mem (hs : convex ℝ s) (hsc : is_closed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
(hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : integrable_on f t μ) :
⨍ x in t, f x ∂μ ∈ s | begin
haveI : fact (μ t < ∞) := ⟨ht.lt_top⟩,
refine hs.average_mem hsc _ hfs hfi,
rwa [ne.def, restrict_eq_zero]
end | lemma | convex.set_average_mem | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"convex",
"fact",
"is_closed"
] | If `μ` is a non-zero finite measure on `α`, `s` is a convex closed set in `E`, and `f` is an
integrable function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `s`:
`⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.set_average_mem_closure (hs : convex ℝ s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞)
(hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s) (hfi : integrable_on f t μ) :
⨍ x in t, f x ∂μ ∈ closure s | hs.closure.set_average_mem is_closed_closure h0 ht (hfs.mono $ λ x hx, subset_closure hx) hfi | lemma | convex.set_average_mem_closure | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"closure",
"convex",
"is_closed_closure",
"subset_closure"
] | If `μ` is a non-zero finite measure on `α`, `s` is a convex set in `E`, and `f` is an integrable
function sending `μ`-a.e. points to `s`, then the average value of `f` belongs to `closure s`:
`⨍ x, f x ∂μ ∈ s`. See also `convex.center_mass_mem` for a finite sum version of this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.average_mem_epigraph [is_finite_measure μ] (hg : convex_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
(⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} | have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2},
from hfs.mono (λ x hx, ⟨hx, le_rfl⟩),
by simpa only [average_pair hfi hgi]
using hg.convex_epigraph.average_mem (hsc.epigraph hgc) hμ ht_mem (hfi.prod_mk hgi) | lemma | convex_on.average_mem_epigraph | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"convex_on",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
concave_on.average_mem_hypograph [is_finite_measure μ] (hg : concave_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
(⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1} | by simpa only [mem_set_of_eq, pi.neg_apply, average_neg, neg_le_neg_iff]
using hg.neg.average_mem_epigraph hgc.neg hsc hμ hfs hfi hgi.neg | lemma | concave_on.average_mem_hypograph | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"concave_on",
"continuous_on",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.map_average_le [is_finite_measure μ] (hg : convex_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
g (⨍ x, f x ∂μ) ≤ ⨍ x, g (f x) ∂μ | (hg.average_mem_epigraph hgc hsc hμ hfs hfi hgi).2 | lemma | convex_on.map_average_le | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"convex_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points to `s`, then the value of `g` at the average value of `f` is less than or equal to
the average value of `g ∘ f` provided ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.le_map_average [is_finite_measure μ] (hg : concave_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hμ : μ ≠ 0) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
⨍ x, g (f x) ∂μ ≤ g (⨍ x, f x ∂μ) | (hg.average_mem_hypograph hgc hsc hμ hfs hfi hgi).2 | lemma | concave_on.le_map_average | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"concave_on",
"continuous_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points to `s`, then the average value of `g ∘ f` is less than or equal to the value of `g`
at the average value of `f` provided... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.set_average_mem_epigraph (hg : convex_on ℝ s g) (hgc : continuous_on g s)
(hsc : is_closed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
(hfi : integrable_on f t μ) (hgi : integrable_on (g ∘ f) t μ) :
(⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} | begin
haveI : fact (μ t < ∞) := ⟨ht.lt_top⟩,
refine hg.average_mem_epigraph hgc hsc _ hfs hfi hgi,
rwa [ne.def, restrict_eq_zero]
end | lemma | convex_on.set_average_mem_epigraph | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"convex_on",
"fact",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points of a set `t` to `s`, then the value of `g` at the average value of `f` over `t` is
less than or equal to the average valu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.set_average_mem_hypograph (hg : concave_on ℝ s g) (hgc : continuous_on g s)
(hsc : is_closed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
(hfi : integrable_on f t μ) (hgi : integrable_on (g ∘ f) t μ) :
(⨍ x in t, f x ∂μ, ⨍ x in t, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ p.2 ≤ g p.1... | by simpa only [mem_set_of_eq, pi.neg_apply, average_neg, neg_le_neg_iff]
using hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg | lemma | concave_on.set_average_mem_hypograph | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"concave_on",
"continuous_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points of a set `t` to `s`, then the average value of `g ∘ f` over `t` is less than or
equal to the value of `g` at the average... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.map_set_average_le (hg : convex_on ℝ s g) (hgc : continuous_on g s)
(hsc : is_closed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
(hfi : integrable_on f t μ) (hgi : integrable_on (g ∘ f) t μ) :
g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ | (hg.set_average_mem_epigraph hgc hsc h0 ht hfs hfi hgi).2 | lemma | convex_on.map_set_average_le | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"convex_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points of a set `t` to `s`, then the value of `g` at the average value of `f` over `t` is
less than or equal to the average valu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.le_map_set_average (hg : concave_on ℝ s g) (hgc : continuous_on g s)
(hsc : is_closed s) (h0 : μ t ≠ 0) (ht : μ t ≠ ∞) (hfs : ∀ᵐ x ∂μ.restrict t, f x ∈ s)
(hfi : integrable_on f t μ) (hgi : integrable_on (g ∘ f) t μ) :
⨍ x in t, g (f x) ∂μ ≤ g (⨍ x in t, f x ∂μ) | (hg.set_average_mem_hypograph hgc hsc h0 ht hfs hfi hgi).2 | lemma | concave_on.le_map_set_average | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"concave_on",
"continuous_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
set `s`, `μ` is a finite non-zero measure on `α`, and `f : α → E` is a function sending
`μ`-a.e. points of a set `t` to `s`, then the average value of `g ∘ f` over `t` is less than or
equal to the value of `g` at the average... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.map_integral_le [is_probability_measure μ] (hg : convex_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ)
(hgi : integrable (g ∘ f) μ) :
g (∫ x, f x ∂μ) ≤ ∫ x, g (f x) ∂μ | by simpa only [average_eq_integral]
using hg.map_average_le hgc hsc (is_probability_measure.ne_zero μ) hfs hfi hgi | lemma | convex_on.map_integral_le | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"convex_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is convex and continuous on a convex closed
set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e. points
to `s`, then the value of `g` at the expected value of `f` is less than or equal to the expected
value of `g ∘ f` provided t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.le_map_integral [is_probability_measure μ] (hg : concave_on ℝ s g)
(hgc : continuous_on g s) (hsc : is_closed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ)
(hgi : integrable (g ∘ f) μ) :
∫ x, g (f x) ∂μ ≤ g (∫ x, f x ∂μ) | by simpa only [average_eq_integral]
using hg.le_map_average hgc hsc (is_probability_measure.ne_zero μ) hfs hfi hgi | lemma | concave_on.le_map_integral | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"concave_on",
"continuous_on",
"is_closed"
] | **Jensen's inequality**: if a function `g : E → ℝ` is concave and continuous on a convex closed
set `s`, `μ` is a probability measure on `α`, and `f : α → E` is a function sending `μ`-a.e. points
to `s`, then the expected value of `g ∘ f` is less than or equal to the value of `g` at the expected
value of `f` provided ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_eq_const_or_exists_average_ne_compl [is_finite_measure μ] (hfi : integrable f μ) :
(f =ᵐ[μ] const α (⨍ x, f x ∂μ)) ∨ ∃ t, measurable_set t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧
⨍ x in t, f x ∂μ ≠ ⨍ x in tᶜ, f x ∂μ | begin
refine or_iff_not_imp_right.mpr (λ H, _), push_neg at H,
refine hfi.ae_eq_of_forall_set_integral_eq _ _ (integrable_const _) (λ t ht ht', _), clear ht',
simp only [const_apply, set_integral_const],
by_cases h₀ : μ t = 0,
{ rw [restrict_eq_zero.2 h₀, integral_zero_measure, h₀, ennreal.zero_to_real, zero_... | lemma | ae_eq_const_or_exists_average_ne_compl | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"ennreal.zero_to_real",
"measurable_set",
"open_segment_same",
"zero_smul"
] | If `f : α → E` is an integrable function, then either it is a.e. equal to the constant
`⨍ x, f x ∂μ` or there exists a measurable set such that `μ t ≠ 0`, `μ tᶜ ≠ 0`, and the average
values of `f` over `t` and `tᶜ` are different. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.average_mem_interior_of_set [is_finite_measure μ] (hs : convex ℝ s) (h0 : μ t ≠ 0)
(hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (ht : ⨍ x in t, f x ∂μ ∈ interior s) :
⨍ x, f x ∂μ ∈ interior s | begin
rw ← measure_to_measurable at h0, rw ← restrict_to_measurable (measure_ne_top μ t) at ht,
by_cases h0' : μ (to_measurable μ t)ᶜ = 0,
{ rw ← ae_eq_univ at h0',
rwa [restrict_congr_set h0', restrict_univ] at ht },
exact hs.open_segment_interior_closure_subset_interior ht
(hs.set_average_mem_closure ... | lemma | convex.average_mem_interior_of_set | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"convex",
"interior"
] | If an integrable function `f : α → E` takes values in a convex set `s` and for some set `t` of
positive measure, the average value of `f` over `t` belongs to the interior of `s`, then the average
of `f` over the whole space belongs to the interior of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex.ae_eq_const_or_average_mem_interior [is_finite_measure μ]
(hs : strict_convex ℝ s) (hsc : is_closed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) :
f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ⨍ x, f x ∂μ ∈ interior s | begin
have : ∀ {t}, μ t ≠ 0 → ⨍ x in t, f x ∂μ ∈ s,
from λ t ht, hs.convex.set_average_mem hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs)
hfi.integrable_on,
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right _,
rintro ⟨t, hm, h₀, h₀', hne⟩,
exact hs.open_segment_subset (this h₀) (this ... | lemma | strict_convex.ae_eq_const_or_average_mem_interior | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"ae_eq_const_or_exists_average_ne_compl",
"interior",
"is_closed",
"strict_convex"
] | If an integrable function `f : α → E` takes values in a strictly convex closed set `s`, then
either it is a.e. equal to its average value, or its average value belongs to the interior of
`s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on.ae_eq_const_or_map_average_lt [is_finite_measure μ]
(hg : strict_convex_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s)
(hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ | begin
have : ∀ {t}, μ t ≠ 0 → ⨍ x in t, f x ∂μ ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ,
from λ t ht, hg.convex_on.set_average_mem_epigraph hgc hsc ht (measure_ne_top _ _)
(ae_restrict_of_ae hfs) hfi.integrable_on hgi.integrable_on,
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right _,... | lemma | strict_convex_on.ae_eq_const_or_map_average_lt | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"ae_eq_const_or_exists_average_ne_compl",
"continuous_on",
"is_closed",
"mul_le_mul_of_nonneg_left",
"prod.mk.inj_iff",
"prod.smul_mk",
"strict_convex_on"
] | **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
convex closed set `s`, and `g : E → ℝ` is continuous and strictly convex on `s`, then
either `f` is a.e. equal to its average value, or `g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on.ae_eq_const_or_lt_map_average [is_finite_measure μ]
(hg : strict_concave_on ℝ s g) (hgc : continuous_on g s) (hsc : is_closed s)
(hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : integrable f μ) (hgi : integrable (g ∘ f) μ) :
f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ) | by simpa only [pi.neg_apply, average_neg, neg_lt_neg_iff]
using hg.neg.ae_eq_const_or_map_average_lt hgc.neg hsc hfs hfi hgi.neg | lemma | strict_concave_on.ae_eq_const_or_lt_map_average | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"continuous_on",
"is_closed",
"strict_concave_on"
] | **Jensen's inequality**, strict version: if an integrable function `f : α → E` takes values in a
convex closed set `s`, and `g : E → ℝ` is continuous and strictly concave on `s`, then
either `f` is a.e. equal to its average value, or `⨍ x, g (f x) ∂μ < g (⨍ x, f x ∂μ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_eq_const_or_norm_average_lt_of_norm_le_const [strict_convex_space ℝ E]
(h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
(f =ᵐ[μ] const α ⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C | begin
cases le_or_lt C 0 with hC0 hC0,
{ have : f =ᵐ[μ] 0, from h_le.mono (λ x hx, norm_le_zero_iff.1 (hx.trans hC0)),
simp only [average_congr this, pi.zero_apply, average_zero],
exact or.inl this },
by_cases hfi : integrable f μ, swap,
by simp [average_eq, integral_undef hfi, hC0, ennreal.to_real_po... | lemma | ae_eq_const_or_norm_average_lt_of_norm_le_const | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"ennreal.to_real_pos_iff",
"interior_closed_ball",
"le_top",
"strict_convex_closed_ball",
"strict_convex_space"
] | If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
a.e., then either this function is a.e. equal to its average value, or the norm of its average value
is strictly less than `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_eq_const_or_norm_integral_lt_of_norm_le_const [strict_convex_space ℝ E]
[is_finite_measure μ] (h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) :
(f =ᵐ[μ] const α ⨍ x, f x ∂μ) ∨ ‖∫ x, f x ∂μ‖ < (μ univ).to_real * C | begin
cases eq_or_ne μ 0 with h₀ h₀, { left, simp [h₀] },
have hμ : 0 < (μ univ).to_real,
by simp [ennreal.to_real_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top],
refine (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right (λ H, _),
rwa [average_eq, norm_smul, norm_inv, real.norm_eq_abs, abs_of_p... | lemma | ae_eq_const_or_norm_integral_lt_of_norm_le_const | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"abs_of_pos",
"ae_eq_const_or_norm_average_lt_of_norm_le_const",
"div_eq_inv_mul",
"div_lt_iff'",
"ennreal.to_real_pos_iff",
"eq_or_ne",
"norm_inv",
"norm_smul",
"real.norm_eq_abs",
"strict_convex_space"
] | If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
a.e., then either this function is a.e. equal to its average value, or the norm of its integral is
strictly less than `(μ univ).to_real * C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ae_eq_const_or_norm_set_integral_lt_of_norm_le_const [strict_convex_space ℝ E]
(ht : μ t ≠ ∞) (h_le : ∀ᵐ x ∂μ.restrict t, ‖f x‖ ≤ C) :
(f =ᵐ[μ.restrict t] const α ⨍ x in t, f x ∂μ) ∨ ‖∫ x in t, f x ∂μ‖ < (μ t).to_real * C | begin
haveI := fact.mk ht.lt_top,
rw [← restrict_apply_univ],
exact ae_eq_const_or_norm_integral_lt_of_norm_le_const h_le
end | lemma | ae_eq_const_or_norm_set_integral_lt_of_norm_le_const | analysis.convex | src/analysis/convex/integral.lean | [
"analysis.convex.function",
"analysis.convex.strict_convex_space",
"measure_theory.function.ae_eq_of_integral",
"measure_theory.integral.average"
] | [
"ae_eq_const_or_norm_integral_lt_of_norm_le_const",
"strict_convex_space"
] | If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
a.e. on a set `t` of finite measure, then either this function is a.e. equal to its average value on
`t`, or the norm of its integral over `t` is strictly less than `(μ t).to_real * C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intrinsic_interior (s : set P) : set P | coe '' interior (coe ⁻¹' s : set $ affine_span 𝕜 s) | def | intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"interior"
] | The intrinsic interior of a set is its interior considered as a set in its affine span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intrinsic_frontier (s : set P) : set P | coe '' frontier (coe ⁻¹' s : set $ affine_span 𝕜 s) | def | intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"frontier"
] | The intrinsic frontier of a set is its frontier considered as a set in its affine span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intrinsic_closure (s : set P) : set P | coe '' closure (coe ⁻¹' s : set $ affine_span 𝕜 s) | def | intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"closure"
] | The intrinsic closure of a set is its closure considered as a set in its affine span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_intrinsic_interior :
x ∈ intrinsic_interior 𝕜 s ↔ ∃ y, y ∈ interior (coe ⁻¹' s : set $ affine_span 𝕜 s) ∧ ↑y = x | mem_image _ _ _ | lemma | mem_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"interior",
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_intrinsic_frontier :
x ∈ intrinsic_frontier 𝕜 s ↔ ∃ y, y ∈ frontier (coe ⁻¹' s : set $ affine_span 𝕜 s) ∧ ↑y = x | mem_image _ _ _ | lemma | mem_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"frontier",
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_intrinsic_closure :
x ∈ intrinsic_closure 𝕜 s ↔ ∃ y, y ∈ closure (coe ⁻¹' s : set $ affine_span 𝕜 s) ∧ ↑y = x | mem_image _ _ _ | lemma | mem_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"closure",
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_interior_subset : intrinsic_interior 𝕜 s ⊆ s | image_subset_iff.2 interior_subset | lemma | intrinsic_interior_subset | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"interior_subset",
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_subset (hs : is_closed s) : intrinsic_frontier 𝕜 s ⊆ s | image_subset_iff.2 (hs.preimage continuous_induced_dom).frontier_subset | lemma | intrinsic_frontier_subset | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"continuous_induced_dom",
"intrinsic_frontier",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_subset_intrinsic_closure :
intrinsic_frontier 𝕜 s ⊆ intrinsic_closure 𝕜 s | image_subset _ frontier_subset_closure | lemma | intrinsic_frontier_subset_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"frontier_subset_closure",
"intrinsic_closure",
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_intrinsic_closure : s ⊆ intrinsic_closure 𝕜 s | λ x hx, ⟨⟨x, subset_affine_span _ _ hx⟩, subset_closure hx, rfl⟩ | lemma | subset_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_closure",
"subset_affine_span",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_interior_empty : intrinsic_interior 𝕜 (∅ : set P) = ∅ | by simp [intrinsic_interior] | lemma | intrinsic_interior_empty | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_empty : intrinsic_frontier 𝕜 (∅ : set P) = ∅ | by simp [intrinsic_frontier] | lemma | intrinsic_frontier_empty | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_empty : intrinsic_closure 𝕜 (∅ : set P) = ∅ | by simp [intrinsic_closure] | lemma | intrinsic_closure_empty | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_nonempty : (intrinsic_closure 𝕜 s).nonempty ↔ s.nonempty | ⟨by { simp_rw nonempty_iff_ne_empty, rintro h rfl, exact h intrinsic_closure_empty },
nonempty.mono subset_intrinsic_closure⟩ | lemma | intrinsic_closure_nonempty | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_closure",
"intrinsic_closure_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_interior_singleton (x : P) : intrinsic_interior 𝕜 ({x} : set P) = {x} | by simpa only [intrinsic_interior, preimage_coe_affine_span_singleton, interior_univ, image_univ,
subtype.range_coe] using coe_affine_span_singleton _ _ _ | lemma | intrinsic_interior_singleton | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"interior_univ",
"intrinsic_interior",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_singleton (x : P) : intrinsic_frontier 𝕜 ({x} : set P) = ∅ | by rw [intrinsic_frontier, preimage_coe_affine_span_singleton, frontier_univ, image_empty] | lemma | intrinsic_frontier_singleton | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"frontier_univ",
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_singleton (x : P) : intrinsic_closure 𝕜 ({x} : set P) = {x} | by simpa only [intrinsic_closure, preimage_coe_affine_span_singleton, closure_univ, image_univ,
subtype.range_coe] using coe_affine_span_singleton _ _ _ | lemma | intrinsic_closure_singleton | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure_univ",
"intrinsic_closure",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_mono (h : s ⊆ t) : intrinsic_closure 𝕜 s ⊆ intrinsic_closure 𝕜 t | begin
refine image_subset_iff.2 (λ x hx, ⟨set.inclusion (affine_span_mono _ h) x,
(continuous_inclusion _).closure_preimage_subset _ $ closure_mono _ hx, rfl⟩),
exact λ y hy, h hy,
end | lemma | intrinsic_closure_mono | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span_mono",
"closure_mono",
"continuous_inclusion",
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_subset_intrinsic_interior : interior s ⊆ intrinsic_interior 𝕜 s | λ x hx, ⟨⟨x, subset_affine_span _ _ $ interior_subset hx⟩,
preimage_interior_subset_interior_preimage continuous_subtype_coe hx, rfl⟩ | lemma | interior_subset_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"continuous_subtype_coe",
"interior",
"interior_subset",
"intrinsic_interior",
"preimage_interior_subset_interior_preimage",
"subset_affine_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_subset_closure : intrinsic_closure 𝕜 s ⊆ closure s | image_subset_iff.2 $ continuous_subtype_coe.closure_preimage_subset _ | lemma | intrinsic_closure_subset_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure",
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_subset_frontier : intrinsic_frontier 𝕜 s ⊆ frontier s | image_subset_iff.2 $ continuous_subtype_coe.frontier_preimage_subset _ | lemma | intrinsic_frontier_subset_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"frontier",
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_subset_affine_span : intrinsic_closure 𝕜 s ⊆ affine_span 𝕜 s | (image_subset_range _ _).trans subtype.range_coe.subset | lemma | intrinsic_closure_subset_affine_span | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_diff_intrinsic_frontier (s : set P) :
intrinsic_closure 𝕜 s \ intrinsic_frontier 𝕜 s = intrinsic_interior 𝕜 s | (image_diff subtype.coe_injective _ _).symm.trans $
by rw [closure_diff_frontier, intrinsic_interior] | lemma | intrinsic_closure_diff_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure_diff_frontier",
"intrinsic_closure",
"intrinsic_frontier",
"intrinsic_interior",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_diff_intrinsic_interior (s : set P) :
intrinsic_closure 𝕜 s \ intrinsic_interior 𝕜 s = intrinsic_frontier 𝕜 s | (image_diff subtype.coe_injective _ _).symm | lemma | intrinsic_closure_diff_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_closure",
"intrinsic_frontier",
"intrinsic_interior",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_interior_union_intrinsic_frontier (s : set P) :
intrinsic_interior 𝕜 s ∪ intrinsic_frontier 𝕜 s = intrinsic_closure 𝕜 s | by simp [intrinsic_closure, intrinsic_interior, intrinsic_frontier,
closure_eq_interior_union_frontier, image_union] | lemma | intrinsic_interior_union_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure_eq_interior_union_frontier",
"intrinsic_closure",
"intrinsic_frontier",
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_frontier_union_intrinsic_interior (s : set P) :
intrinsic_frontier 𝕜 s ∪ intrinsic_interior 𝕜 s = intrinsic_closure 𝕜 s | by rw [union_comm, intrinsic_interior_union_intrinsic_frontier] | lemma | intrinsic_frontier_union_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"intrinsic_closure",
"intrinsic_frontier",
"intrinsic_interior",
"intrinsic_interior_union_intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_intrinsic_closure (hs : is_closed (affine_span 𝕜 s : set P)) :
is_closed (intrinsic_closure 𝕜 s) | (closed_embedding_subtype_coe hs).is_closed_map _ is_closed_closure | lemma | is_closed_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"closed_embedding_subtype_coe",
"intrinsic_closure",
"is_closed",
"is_closed_closure",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_intrinsic_frontier (hs : is_closed (affine_span 𝕜 s : set P)) :
is_closed (intrinsic_frontier 𝕜 s) | (closed_embedding_subtype_coe hs).is_closed_map _ is_closed_frontier | lemma | is_closed_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"closed_embedding_subtype_coe",
"intrinsic_frontier",
"is_closed",
"is_closed_frontier",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
affine_span_intrinsic_closure (s : set P) :
affine_span 𝕜 (intrinsic_closure 𝕜 s) = affine_span 𝕜 s | (affine_span_le.2 intrinsic_closure_subset_affine_span).antisymm $
affine_span_mono _ subset_intrinsic_closure | lemma | affine_span_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"affine_span_mono",
"intrinsic_closure",
"intrinsic_closure_subset_affine_span",
"subset_intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.intrinsic_closure (hs : is_closed (coe ⁻¹' s : set $ affine_span 𝕜 s)) :
intrinsic_closure 𝕜 s = s | begin
rw [intrinsic_closure, hs.closure_eq, image_preimage_eq_of_subset],
exact (subset_affine_span _ _).trans subtype.range_coe.superset,
end | lemma | is_closed.intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"intrinsic_closure",
"is_closed",
"subset_affine_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_idem (s : set P) :
intrinsic_closure 𝕜 (intrinsic_closure 𝕜 s) = intrinsic_closure 𝕜 s | begin
refine is_closed.intrinsic_closure _,
set t := affine_span 𝕜 (intrinsic_closure 𝕜 s) with ht,
clear_value t,
obtain rfl := ht.trans (affine_span_intrinsic_closure _),
rw [intrinsic_closure, preimage_image_eq _ subtype.coe_injective],
exact is_closed_closure,
end | lemma | intrinsic_closure_idem | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"affine_span_intrinsic_closure",
"intrinsic_closure",
"is_closed.intrinsic_closure",
"is_closed_closure",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_intrinsic_interior (φ : P →ᵃⁱ[𝕜] Q) (s : set P) :
intrinsic_interior 𝕜 (φ '' s) = φ '' intrinsic_interior 𝕜 s | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp only [intrinsic_interior_empty, image_empty] },
haveI : nonempty s := hs.to_subtype,
let f := ((affine_span 𝕜 s).isometry_equiv_map φ).to_homeomorph,
have : φ.to_affine_map ∘ coe ∘ f.symm = coe := funext isometry_equiv_map.apply_symm_apply,
rw [intrin... | lemma | affine_isometry.image_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_isometry.coe_to_affine_map",
"affine_span",
"intrinsic_interior",
"intrinsic_interior_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_intrinsic_frontier (φ : P →ᵃⁱ[𝕜] Q) (s : set P) :
intrinsic_frontier 𝕜 (φ '' s) = φ '' intrinsic_frontier 𝕜 s | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp },
haveI : nonempty s := hs.to_subtype,
let f := ((affine_span 𝕜 s).isometry_equiv_map φ).to_homeomorph,
have : φ.to_affine_map ∘ coe ∘ f.symm = coe := funext isometry_equiv_map.apply_symm_apply,
rw [intrinsic_frontier, intrinsic_frontier, ←φ.coe_to_a... | lemma | affine_isometry.image_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_isometry.coe_to_affine_map",
"affine_span",
"intrinsic_frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_intrinsic_closure (φ : P →ᵃⁱ[𝕜] Q) (s : set P) :
intrinsic_closure 𝕜 (φ '' s) = φ '' intrinsic_closure 𝕜 s | begin
obtain rfl | hs := s.eq_empty_or_nonempty,
{ simp },
haveI : nonempty s := hs.to_subtype,
let f := ((affine_span 𝕜 s).isometry_equiv_map φ).to_homeomorph,
have : φ.to_affine_map ∘ coe ∘ f.symm = coe := funext isometry_equiv_map.apply_symm_apply,
rw [intrinsic_closure, intrinsic_closure, ←φ.coe_to_aff... | lemma | affine_isometry.image_intrinsic_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_isometry.coe_to_affine_map",
"affine_span",
"intrinsic_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
intrinsic_closure_eq_closure : intrinsic_closure 𝕜 s = closure s | begin
ext x,
simp only [mem_closure_iff, mem_intrinsic_closure],
refine ⟨_, λ h, ⟨⟨x, _⟩, _, subtype.coe_mk _ _⟩⟩,
{ rintro ⟨x, h, rfl⟩ t ht hx,
obtain ⟨z, hz₁, hz₂⟩ := h _ (continuous_induced_dom.is_open_preimage t ht) hx,
exact ⟨z, hz₁, hz₂⟩ },
{ by_contradiction hc,
obtain ⟨z, hz₁, hz₂⟩ := h _ ... | lemma | intrinsic_closure_eq_closure | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_span",
"by_contradiction",
"closure",
"intrinsic_closure",
"mem_closure_iff",
"mem_intrinsic_closure",
"subset_affine_span",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_diff_intrinsic_interior (s : set P) :
closure s \ intrinsic_interior 𝕜 s = intrinsic_frontier 𝕜 s | intrinsic_closure_eq_closure 𝕜 s ▸ intrinsic_closure_diff_intrinsic_interior s | lemma | closure_diff_intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure",
"intrinsic_closure_diff_intrinsic_interior",
"intrinsic_closure_eq_closure",
"intrinsic_frontier",
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_diff_intrinsic_frontier (s : set P) :
closure s \ intrinsic_frontier 𝕜 s = intrinsic_interior 𝕜 s | intrinsic_closure_eq_closure 𝕜 s ▸ intrinsic_closure_diff_intrinsic_frontier s | lemma | closure_diff_intrinsic_frontier | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"closure",
"intrinsic_closure_diff_intrinsic_frontier",
"intrinsic_closure_eq_closure",
"intrinsic_frontier",
"intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aux {α β : Type*} [topological_space α] [topological_space β] (φ : α ≃ₜ β)
(s : set β) :
(interior s).nonempty ↔ (interior (φ ⁻¹' s)).nonempty | by rw [←φ.image_symm, ←φ.symm.image_interior, nonempty_image_iff] | lemma | aux | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"interior",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.nonempty.intrinsic_interior (hscv : convex ℝ s) (hsne : s.nonempty) :
(intrinsic_interior ℝ s).nonempty | begin
haveI := hsne.coe_sort,
obtain ⟨p, hp⟩ := hsne,
let p' : affine_span ℝ s := ⟨p, subset_affine_span _ _ hp⟩,
rw [intrinsic_interior, nonempty_image_iff,
aux (affine_isometry_equiv.const_vsub ℝ p').symm.to_homeomorph,
convex.interior_nonempty_iff_affine_span_eq_top, affine_isometry_equiv.coe_to_home... | lemma | set.nonempty.intrinsic_interior | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"affine_isometry_equiv.coe_to_homeomorph",
"affine_isometry_equiv.const_vsub",
"affine_span",
"affine_span_coe_preimage_eq_top",
"aux",
"convex",
"convex.interior_nonempty_iff_affine_span_eq_top",
"intrinsic_interior",
"subset_affine_span"
] | The intrinsic interior of a nonempty convex set is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
intrinsic_interior_nonempty (hs : convex ℝ s) :
(intrinsic_interior ℝ s).nonempty ↔ s.nonempty | ⟨by { simp_rw nonempty_iff_ne_empty, rintro h rfl, exact h intrinsic_interior_empty },
set.nonempty.intrinsic_interior hs⟩ | lemma | intrinsic_interior_nonempty | analysis.convex | src/analysis/convex/intrinsic.lean | [
"analysis.normed_space.add_torsor_bases"
] | [
"convex",
"intrinsic_interior",
"intrinsic_interior_empty",
"set.nonempty.intrinsic_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.map_center_mass_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : 0 < ∑ i in t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
f (t.center_mass w p) ≤ t.center_mass w (f ∘ p) | begin
have hmem' : ∀ i ∈ t, (p i, (f ∘ p) i) ∈ {p : E × β | p.1 ∈ s ∧ f p.1 ≤ p.2},
from λ i hi, ⟨hmem i hi, le_rfl⟩,
convert (hf.convex_epigraph.center_mass_mem h₀ h₁ hmem').2;
simp only [center_mass, function.comp, prod.smul_fst, prod.fst_sum,
prod.smul_snd, prod.snd_sum],
end | lemma | convex_on.map_center_mass_le | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"convex_on",
"prod.smul_fst",
"prod.smul_snd"
] | Convex **Jensen's inequality**, `finset.center_mass` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.le_map_center_mass (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : 0 < ∑ i in t, w i) (hmem : ∀ i ∈ t, p i ∈ s) :
t.center_mass w (f ∘ p) ≤ f (t.center_mass w p) | @convex_on.map_center_mass_le 𝕜 E βᵒᵈ _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem | lemma | concave_on.le_map_center_mass | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"concave_on",
"convex_on.map_center_mass_le"
] | Concave **Jensen's inequality**, `finset.center_mass` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.map_sum_le (hf : convex_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
(hmem : ∀ i ∈ t, p i ∈ s) :
f (∑ i in t, w i • p i) ≤ ∑ i in t, w i • f (p i) | by simpa only [center_mass, h₁, inv_one, one_smul]
using hf.map_center_mass_le h₀ (h₁.symm ▸ zero_lt_one) hmem | lemma | convex_on.map_sum_le | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"convex_on",
"inv_one",
"one_smul",
"zero_lt_one"
] | Convex **Jensen's inequality**, `finset.sum` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.le_map_sum (hf : concave_on 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i in t, w i = 1)
(hmem : ∀ i ∈ t, p i ∈ s) :
∑ i in t, w i • f (p i) ≤ f (∑ i in t, w i • p i) | @convex_on.map_sum_le 𝕜 E βᵒᵈ _ _ _ _ _ _ _ _ _ _ _ _ hf h₀ h₁ hmem | lemma | concave_on.le_map_sum | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"concave_on",
"convex_on.map_sum_le"
] | Concave **Jensen's inequality**, `finset.sum` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sup_of_mem_convex_hull {s : finset E} (hf : convex_on 𝕜 (convex_hull 𝕜 (s : set E)) f)
(hx : x ∈ convex_hull 𝕜 (s : set E)) :
f x ≤ s.sup' (coe_nonempty.1 $ convex_hull_nonempty_iff.1 ⟨x, hx⟩) f | begin
obtain ⟨w, hw₀, hw₁, rfl⟩ := mem_convex_hull.1 hx,
exact (hf.map_center_mass_le hw₀ (by positivity) $ subset_convex_hull _ _).trans
(center_mass_le_sup hw₀ $ by positivity),
end | lemma | le_sup_of_mem_convex_hull | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"convex_hull",
"convex_on",
"finset",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_le_of_mem_convex_hull {s : finset E} (hf : concave_on 𝕜 (convex_hull 𝕜 (s : set E)) f)
(hx : x ∈ convex_hull 𝕜 (s : set E)) :
s.inf' (coe_nonempty.1 $ convex_hull_nonempty_iff.1 ⟨x, hx⟩) f ≤ f x | le_sup_of_mem_convex_hull hf.dual hx | lemma | inf_le_of_mem_convex_hull | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"concave_on",
"convex_hull",
"finset",
"le_sup_of_mem_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on.exists_ge_of_center_mass (h : convex_on 𝕜 s f)
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i in t, w i) (hp : ∀ i ∈ t, p i ∈ s) :
∃ i ∈ t, f (t.center_mass w p) ≤ f (p i) | begin
set y := t.center_mass w p,
rsuffices ⟨i, hi, hfi⟩ : ∃ i ∈ t.filter (λ i, w i ≠ 0), w i • f y ≤ w i • (f ∘ p) i,
{ rw mem_filter at hi,
exact ⟨i, hi.1, (smul_le_smul_iff_of_pos $ (hw₀ i hi.1).lt_of_ne hi.2.symm).1 hfi⟩ },
have hw' : (0 : 𝕜) < ∑ i in filter (λ i, w i ≠ 0) t, w i := by rwa sum_filter_n... | lemma | convex_on.exists_ge_of_center_mass | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"convex_on",
"filter",
"finset.center_mass_filter_ne_zero",
"inv_smul_smul₀",
"smul_le_smul_iff_of_pos"
] | If a function `f` is convex on `s`, then the value it takes at some center of mass of points of
`s` is less than the value it takes on one of those points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.exists_le_of_center_mass (h : concave_on 𝕜 s f)
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : 0 < ∑ i in t, w i) (hp : ∀ i ∈ t, p i ∈ s) :
∃ i ∈ t, f (p i) ≤ f (t.center_mass w p) | @convex_on.exists_ge_of_center_mass 𝕜 E βᵒᵈ _ _ _ _ _ _ _ _ _ _ _ _ h hw₀ hw₁ hp | lemma | concave_on.exists_le_of_center_mass | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"concave_on",
"convex_on.exists_ge_of_center_mass"
] | If a function `f` is concave on `s`, then the value it takes at some center of mass of points of
`s` is greater than the value it takes on one of those points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on.exists_ge_of_mem_convex_hull (hf : convex_on 𝕜 (convex_hull 𝕜 s) f) {x}
(hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f x ≤ f y | begin
rw _root_.convex_hull_eq at hx,
obtain ⟨α, t, w, p, hw₀, hw₁, hp, rfl⟩ := hx,
rcases hf.exists_ge_of_center_mass hw₀ (hw₁.symm ▸ zero_lt_one)
(λ i hi, subset_convex_hull 𝕜 s (hp i hi)) with ⟨i, hit, Hi⟩,
exact ⟨p i, hp i hit, Hi⟩
end | lemma | convex_on.exists_ge_of_mem_convex_hull | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"convex_hull",
"convex_on",
"subset_convex_hull",
"zero_lt_one"
] | Maximum principle for convex functions. If a function `f` is convex on the convex hull of `s`,
then the eventual maximum of `f` on `convex_hull 𝕜 s` lies in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on.exists_le_of_mem_convex_hull (hf : concave_on 𝕜 (convex_hull 𝕜 s) f) {x}
(hx : x ∈ convex_hull 𝕜 s) : ∃ y ∈ s, f y ≤ f x | @convex_on.exists_ge_of_mem_convex_hull 𝕜 E βᵒᵈ _ _ _ _ _ _ _ _ hf _ hx | lemma | concave_on.exists_le_of_mem_convex_hull | analysis.convex | src/analysis/convex/jensen.lean | [
"analysis.convex.combination",
"analysis.convex.function"
] | [
"concave_on",
"convex_hull",
"convex_on.exists_ge_of_mem_convex_hull"
] | Minimum principle for concave functions. If a function `f` is concave on the convex hull of `s`,
then the eventual minimum of `f` on `convex_hull 𝕜 s` lies in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_join (s t : set E) : set E | ⋃ (x ∈ s) (y ∈ t), segment 𝕜 x y | def | convex_join | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"segment"
] | The join of two sets is the union of the segments joining them. This can be interpreted as the
topological join, but within the original space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_convex_join : x ∈ convex_join 𝕜 s t ↔ ∃ (a ∈ s) (b ∈ t), x ∈ segment 𝕜 a b | by simp [convex_join] | lemma | mem_convex_join | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_comm (s t : set E) : convex_join 𝕜 s t = convex_join 𝕜 t s | (Union₂_comm _).trans $ by simp_rw [convex_join, segment_symm] | lemma | convex_join_comm | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : convex_join 𝕜 s₁ t₁ ⊆ convex_join 𝕜 s₂ t₂ | bUnion_mono hs $ λ x hx, bUnion_mono ht $ λ y hy, subset.rfl | lemma | convex_join_mono | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_mono_left (hs : s₁ ⊆ s₂) : convex_join 𝕜 s₁ t ⊆ convex_join 𝕜 s₂ t | convex_join_mono hs subset.rfl | lemma | convex_join_mono_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_mono_right (ht : t₁ ⊆ t₂) : convex_join 𝕜 s t₁ ⊆ convex_join 𝕜 s t₂ | convex_join_mono subset.rfl ht | lemma | convex_join_mono_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_empty_left (t : set E) : convex_join 𝕜 ∅ t = ∅ | by simp [convex_join] | lemma | convex_join_empty_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_empty_right (s : set E) : convex_join 𝕜 s ∅ = ∅ | by simp [convex_join] | lemma | convex_join_empty_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_singleton_left (t : set E) (x : E) :
convex_join 𝕜 {x} t = ⋃ (y ∈ t), segment 𝕜 x y | by simp [convex_join] | lemma | convex_join_singleton_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_singleton_right (s : set E) (y : E) :
convex_join 𝕜 s {y} = ⋃ (x ∈ s), segment 𝕜 x y | by simp [convex_join] | lemma | convex_join_singleton_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_singletons (x : E) : convex_join 𝕜 {x} {y} = segment 𝕜 x y | by simp [convex_join] | lemma | convex_join_singletons | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_union_left (s₁ s₂ t : set E) :
convex_join 𝕜 (s₁ ∪ s₂) t = convex_join 𝕜 s₁ t ∪ convex_join 𝕜 s₂ t | by simp_rw [convex_join, mem_union, Union_or, Union_union_distrib] | lemma | convex_join_union_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_union_right (s t₁ t₂ : set E) :
convex_join 𝕜 s (t₁ ∪ t₂) = convex_join 𝕜 s t₁ ∪ convex_join 𝕜 s t₂ | by simp_rw [convex_join, mem_union, Union_or, Union_union_distrib] | lemma | convex_join_union_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_Union_left (s : ι → set E) (t : set E) :
convex_join 𝕜 (⋃ i, s i) t = ⋃ i, convex_join 𝕜 (s i) t | by { simp_rw [convex_join, mem_Union, Union_exists], exact Union_comm _ } | lemma | convex_join_Union_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_Union_right (s : set E) (t : ι → set E) :
convex_join 𝕜 s (⋃ i, t i) = ⋃ i, convex_join 𝕜 s (t i) | by simp_rw [convex_join_comm s, convex_join_Union_left] | lemma | convex_join_Union_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_Union_left",
"convex_join_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
segment_subset_convex_join (hx : x ∈ s) (hy : y ∈ t) : segment 𝕜 x y ⊆ convex_join 𝕜 s t | (subset_Union₂ y hy).trans (subset_Union₂ x hx) | lemma | segment_subset_convex_join | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_convex_join_left (h : t.nonempty) : s ⊆ convex_join 𝕜 s t | λ x hx, let ⟨y, hy⟩ := h in segment_subset_convex_join hx hy $ left_mem_segment _ _ _ | lemma | subset_convex_join_left | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"left_mem_segment",
"segment_subset_convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_convex_join_right (h : s.nonempty) : t ⊆ convex_join 𝕜 s t | λ y hy, let ⟨x, hx⟩ := h in segment_subset_convex_join hx hy $ right_mem_segment _ _ _ | lemma | subset_convex_join_right | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"right_mem_segment",
"segment_subset_convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_subset (hs : s ⊆ u) (ht : t ⊆ u) (hu : convex 𝕜 u) : convex_join 𝕜 s t ⊆ u | Union₂_subset $ λ x hx, Union₂_subset $ λ y hy, hu.segment_subset (hs hx) (ht hy) | lemma | convex_join_subset | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex",
"convex_join"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_subset_convex_hull (s t : set E) : convex_join 𝕜 s t ⊆ convex_hull 𝕜 (s ∪ t) | convex_join_subset ((subset_union_left _ _).trans $ subset_convex_hull _ _)
((subset_union_right _ _).trans $ subset_convex_hull _ _) $ convex_convex_hull _ _ | lemma | convex_join_subset_convex_hull | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_convex_hull",
"convex_hull",
"convex_join",
"convex_join_subset",
"subset_convex_hull"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_assoc_aux (s t u : set E) :
convex_join 𝕜 (convex_join 𝕜 s t) u ⊆ convex_join 𝕜 s (convex_join 𝕜 t u) | begin
simp_rw [subset_def, mem_convex_join],
rintro _ ⟨z, ⟨x, hx, y, hy, a₁, b₁, ha₁, hb₁, hab₁, rfl⟩, z, hz, a₂, b₂, ha₂, hb₂, hab₂, rfl⟩,
obtain rfl | hb₂ := hb₂.eq_or_lt,
{ refine ⟨x, hx, y, ⟨y, hy, z, hz, left_mem_segment _ _ _⟩, a₁, b₁, ha₁, hb₁, hab₁, _⟩,
rw add_zero at hab₂,
rw [hab₂, one_smul, z... | lemma | convex_join_assoc_aux | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"div_nonneg",
"div_self",
"left_mem_segment",
"mem_convex_join",
"mul_div_cancel'",
"mul_one",
"one_smul",
"smul_add",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_assoc (s t u : set E) :
convex_join 𝕜 (convex_join 𝕜 s t) u = convex_join 𝕜 s (convex_join 𝕜 t u) | begin
refine (convex_join_assoc_aux _ _ _).antisymm _,
simp_rw [convex_join_comm s, convex_join_comm _ u],
exact convex_join_assoc_aux _ _ _,
end | lemma | convex_join_assoc | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_assoc_aux",
"convex_join_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_left_comm (s t u : set E) :
convex_join 𝕜 s (convex_join 𝕜 t u) = convex_join 𝕜 t (convex_join 𝕜 s u) | by simp_rw [←convex_join_assoc, convex_join_comm] | lemma | convex_join_left_comm | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_right_comm (s t u : set E) :
convex_join 𝕜 (convex_join 𝕜 s t) u = convex_join 𝕜 (convex_join 𝕜 s u) t | by simp_rw [convex_join_assoc, convex_join_comm] | lemma | convex_join_right_comm | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_assoc",
"convex_join_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_convex_join_convex_join_comm (s t u v : set E) :
convex_join 𝕜 (convex_join 𝕜 s t) (convex_join 𝕜 u v) =
convex_join 𝕜 (convex_join 𝕜 s u) (convex_join 𝕜 t v) | by simp_rw [←convex_join_assoc, convex_join_right_comm] | lemma | convex_join_convex_join_convex_join_comm | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_join",
"convex_join_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_hull_insert (hs : s.nonempty) :
convex_hull 𝕜 (insert x s) = convex_join 𝕜 {x} (convex_hull 𝕜 s) | begin
classical,
refine (convex_join_subset ((singleton_subset_iff.2 $ mem_insert _ _).trans $ subset_convex_hull
_ _) (convex_hull_mono $ subset_insert _ _) $ convex_convex_hull _ _).antisymm' (λ x hx, _),
rw convex_hull_eq at hx,
obtain ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩ := hx,
have : (∑ i in t.filter (λ i... | lemma | convex_hull_insert | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"antisymm'",
"convex_convex_hull",
"convex_hull",
"convex_hull_eq",
"convex_hull_mono",
"convex_join",
"convex_join_subset",
"finset.center_mass",
"finset.center_mass_eq_of_sum_1",
"finset.center_mass_mem_convex_hull",
"finset.filter_subset",
"finset.mem_filter",
"finset.sum_smul",
"one_sm... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_segments (a b c d : E) :
convex_join 𝕜 (segment 𝕜 a b) (segment 𝕜 c d) = convex_hull 𝕜 {a, b, c, d} | by simp only [convex_hull_insert, insert_nonempty, singleton_nonempty, convex_hull_pair,
←convex_join_assoc, convex_join_singletons] | lemma | convex_join_segments | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_hull",
"convex_hull_insert",
"convex_hull_pair",
"convex_join",
"convex_join_singletons",
"segment"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_join_segment_singleton (a b c : E) :
convex_join 𝕜 (segment 𝕜 a b) {c} = convex_hull 𝕜 {a, b, c} | by rw [←pair_eq_singleton, ←convex_join_segments, segment_same, pair_eq_singleton] | lemma | convex_join_segment_singleton | analysis.convex | src/analysis/convex/join.lean | [
"analysis.convex.combination"
] | [
"convex_hull",
"convex_join",
"segment",
"segment_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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