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namespace OAI
universe uE uX uY
noncomputable section
open scoped BigOperators NNReal ContDiff ENNReal
open Filter MeasureTheory Set
namespace RadialTransition
section CompactEnvelope
variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
theorem support_norm_bound {F : E → ℝ} {D : Set E} {x v : E} {r M : ℝ}
(hr : 0 < r) (hM : 0 ≤ M) (hball : Metric.closedBall x r ⊆ D)
(hbound : ∀ y ∈ Metric.closedBall x r, |F y| ≤ M)
(hv : SupportsOn F D x v) : ‖v‖ ≤ 2 * M / r := by
apply norm_le_of_unit_inner_le _ (by positivity)
intro z hz
have hy : x + r • z ∈ Metric.closedBall x r := by
simp [Metric.mem_closedBall, dist_eq_norm, norm_smul, abs_of_pos hr, hz]
have hx : x ∈ Metric.closedBall x r := Metric.mem_closedBall_self hr.le
have hh := hv (x + r • z) (hball hy)
simp only [add_sub_cancel_left, real_inner_smul_right] at hh
have hlo := (abs_le.mp (hbound x hx)).1
have hhi := (abs_le.mp (hbound _ hy)).2
apply (le_div_iff₀ hr).mpr
nlinarith
theorem supportsOn_of_tendsto {F : ℕ → E → ℝ} {F₀ : E → ℝ} {D : Set E}
{x v : ℕ → E} {x₀ v₀ : E}
(hx : Tendsto x atTop (nhds x₀)) (hv : Tendsto v atTop (nhds v₀))
(hbase : Tendsto (fun n => F n (x n)) atTop (nhds (F₀ x₀)))
(hpoint : ∀ y ∈ D, Tendsto (fun n => F n y) atTop (nhds (F₀ y)))
(hs : ∀ n, SupportsOn (F n) D (x n) (v n)) :
SupportsOn F₀ D x₀ v₀ := by
intro y hy
exact le_of_tendsto_of_tendsto
(hbase.add (hv.inner (tendsto_const_nhds.sub hx))) (hpoint y hy)
(Eventually.of_forall fun n => hs n y hy)
theorem continuousOn_tendsto_sequence {X : Type uX} {Y : Type uY} [TopologicalSpace X]
[TopologicalSpace Y] {f : X → Y} {S : Set X} (hf : ContinuousOn f S)
{a : X} (ha : a ∈ S) {s : ℕ → X} (hs : ∀ n, s n ∈ S)
(hsa : Tendsto s atTop (nhds a)) : Tendsto (fun n => f (s n)) atTop (nhds (f a)) := by
apply (hf.continuousWithinAt ha).tendsto.comp
exact tendsto_nhdsWithin_iff.mpr ⟨hsa, Eventually.of_forall hs⟩
theorem supportSet_envelope_closedHull [FiniteDimensional ℝ E]
{F : ℝ → E → ℝ} {P : E → ℝ} {D : Set E} {K : Set ℝ} {x : E} {r : ℝ}
(hD : IsOpen D) (hK : IsCompact K) (hr : 0 < r)
(hball : Metric.closedBall x (2 * r) ⊆ D)
(hF : ∀ i ∈ K, ConvexOn ℝ D (F i))
(hFc : ContinuousOn (fun z : ℝ × E => F z.1 z.2) (K ×ˢ D))
(hPc : ContinuousOn P D)
(hle : ∀ i ∈ K, ∀ y ∈ D, F i y ≤ P y)
(hactive : ∀ y ∈ Metric.closedBall x r, ∃ i ∈ K, F i y = P y) :
supportSet P D x = closure (convexHull ℝ
{v | ∃ i ∈ K, F i x = P x ∧ SupportsOn (F i) D x v}) := by
classical
let G : Set E := {v | ∃ i ∈ K, F i x = P x ∧ SupportsOn (F i) D x v}
have hx : x ∈ D := hball (Metric.mem_closedBall_self (by positivity))
have hG : G ⊆ supportSet P D x := by
rintro v ⟨i, hi, heq, hv⟩ y hy
rw [← heq]
exact (hv y hy).trans (hle i hi y hy)
have hback : closure (convexHull ℝ G) ⊆ supportSet P D x :=
closure_minimal (convexHull_min hG (convex_supportSet P D x))
(isClosed_supportSet P D x)
apply Set.Subset.antisymm ?_ hback
intro v hv
by_contra hvout
obtain ⟨ell, c, hsep, hvc⟩ := geometric_hahn_banach_closed_point
(convex_convexHull ℝ G).closure isClosed_closure hvout
let z : E := (InnerProductSpace.toDual ℝ E).symm ell
have hell (w : E) : inner ℝ w z = ell w := by
rw [real_inner_comm]
exact InnerProductSpace.toDual_symm_apply
obtain ⟨M₀, hM₀⟩ := (hK.prod (isCompact_closedBall x (2 * r))).exists_bound_of_continuousOn
(hFc.mono (Set.prod_mono Subset.rfl hball))
let M := max 0 M₀
have hM : 0 ≤ M := le_max_left _ _
have hbound (i : ℝ) (hi : i ∈ K) (y : E) (hy : y ∈ Metric.closedBall x (2 * r)) :
|F i y| ≤ M := (hM₀ (i, y) ⟨hi, hy⟩).trans (le_max_right _ _)
let s : ℕ → ℝ := fun n => (r / (‖z‖ + 1)) * (1 / ((n : ℝ) + 1))
have hspos (n : ℕ) : 0 < s n := by dsimp [s]; positivity
have hsbound (n : ℕ) : s n * ‖z‖ ≤ r := by
have hn : 1 ≤ (n : ℝ) + 1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith
have hden : 0 < ‖z‖ + 1 := by positivity
have hdiv : 1 / ((n : ℝ) + 1) ≤ 1 := (div_le_one (by positivity)).mpr hn
have hq : r / (‖z‖ + 1) * ‖z‖ ≤ r := by
rw [div_mul_eq_mul_div]
exact (div_le_iff₀ hden).mpr (show r * ‖z‖ ≤ r * (‖z‖ + 1) by nlinarith)
calc
s n * ‖z‖ = (r / (‖z‖ + 1) * ‖z‖) * (1 / ((n : ℝ) + 1)) := by dsimp [s]; ring
_ ≤ (r / (‖z‖ + 1) * ‖z‖) * 1 :=
mul_le_mul_of_nonneg_left hdiv (by positivity)
_ ≤ r := by simpa using hq
have hs0 : Tendsto s atTop (nhds 0) := by
simpa [s] using tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (r / (‖z‖ + 1))
let y : ℕ → E := fun n => x + s n • z
have hymem (n : ℕ) : y n ∈ Metric.closedBall x r := by
simpa [y, Metric.mem_closedBall, dist_eq_norm, norm_smul, abs_of_pos (hspos n)] using hsbound n
have hyD (n : ℕ) : y n ∈ D :=
hball (Metric.closedBall_subset_closedBall (by linarith) (hymem n))
have hy0 : Tendsto y atTop (nhds x) := by
simpa [y] using tendsto_const_nhds.add (hs0.smul (tendsto_const_nhds (x := z)))
choose i hi hia using fun n => hactive (y n) (hymem n)
choose w hw using fun n => supportSet_nonempty hD (hF (i n) (hi n)) (hyD n)
have hwbound (n : ℕ) : ‖w n‖ ≤ 2 * M / r := by
have hb : Metric.closedBall (y n) r ⊆ Metric.closedBall x (2 * r) := by
intro q hq
apply (dist_triangle q (y n) x).trans
have hqn := Metric.mem_closedBall.mp hq
have hyn := Metric.mem_closedBall.mp (hymem n)
linarith
exact support_norm_bound hr hM (hb.trans hball)
(fun q hq => hbound (i n) (hi n) q (hb hq)) (hw n)
obtain ⟨iw, hiw, σ, hσ, hlim⟩ :=
(hK.prod (isCompact_closedBall (0 : E) (2 * M / r))).tendsto_subseq
(x := fun n => (i n, w n)) (fun n => ⟨hi n, by simpa using hwbound n⟩)
have hiwK : iw.1 ∈ K := hiw.1
have hilim : Tendsto (fun n => i (σ n)) atTop (nhds iw.1) :=
continuous_fst.tendsto iw |>.comp hlim
have hwlim : Tendsto (fun n => w (σ n)) atTop (nhds iw.2) :=
continuous_snd.tendsto iw |>.comp hlim
have hylim : Tendsto (fun n => y (σ n)) atTop (nhds x) := hy0.comp hσ.tendsto_atTop
have hbase : Tendsto (fun n => F (i (σ n)) (y (σ n))) atTop (nhds (F iw.1 x)) :=
continuousOn_tendsto_sequence hFc (a := (iw.1, x)) ⟨hiwK, hx⟩
(fun n => ⟨hi (σ n), hyD (σ n)⟩) (hilim.prodMk_nhds hylim)
have hPbase : Tendsto (fun n => P (y (σ n))) atTop (nhds (P x)) :=
continuousOn_tendsto_sequence hPc hx (fun n => hyD (σ n)) hylim
have hia0 : F iw.1 x = P x := by
apply tendsto_nhds_unique hbase
simpa only [hia] using hPbase
have hiws : SupportsOn (F iw.1) D x iw.2 := by
apply supportsOn_of_tendsto hylim hwlim hbase ?_ (fun n => hw (σ n))
intro q hq
exact continuousOn_tendsto_sequence hFc (a := (iw.1, q)) ⟨hiwK, hq⟩
(fun n => ⟨hi (σ n), hq⟩) (hilim.prodMk_nhds tendsto_const_nhds)
have hiwG : iw.2 ∈ G := ⟨iw.1, hiwK, hia0, hiws⟩
have hendpoint (n : ℕ) : inner ℝ v z ≤ inner ℝ (w n) z := by
have hleft := hv (y n) (hyD n)
have hright := hw n x hx
have hdom := hle (i n) (hi n) x hx
have hyx : y n - x = s n • z := by simp [y]
have hxy : x - y n = -(s n • z) := by simp [y]
rw [hyx, real_inner_smul_right] at hleft
rw [hxy, inner_neg_right, real_inner_smul_right, hia n] at hright
exact (mul_le_mul_iff_right₀ (hspos n)).mp (by nlinarith [hleft, hright, hdom])
have hfinal : inner ℝ v z ≤ inner ℝ iw.2 z :=
le_of_tendsto_of_tendsto tendsto_const_nhds (hwlim.inner tendsto_const_nhds)
(Eventually.of_forall fun n => hendpoint (σ n))
rw [hell, hell] at hfinal
have hh := hsep iw.2 (subset_closure (subset_convexHull ℝ G hiwG))
linarith
end CompactEnvelope
end RadialTransition
end
end OAI
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