| import OAI.Analysis.DiamondDistortion.AMUC |
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| namespace OAI |
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| universe u v |
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| noncomputable section |
| open scoped BigOperators Classical Topology |
| open Set |
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| namespace DiamondDistortion |
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| namespace Midpoint |
| variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| / |
| convex combinations. -/ |
| lemma lens_shrink {a R : ℝ} (ha : 0 ≤ a) (ha1 : a ≤ 1) (u v : X) |
| (hp : ‖u+v‖ ≤ R) (hm : ‖u-v‖ ≤ R) : |
| max ‖u+a • v‖ ‖u-a • v‖ ≤ R := by |
| have hn1 : 0 ≤ (1+a)/2 := by positivity |
| have hn2 : 0 ≤ (1-a)/2 := by linarith |
| have hn (v w : X) (hv : ‖v‖ ≤ R) (hw : ‖w‖ ≤ R) : |
| ‖((1+a)/2) • v+((1-a)/2) • w‖ ≤ R := by |
| calc |
| _ ≤ ‖((1+a)/2) • v‖+‖((1-a)/2) • w‖ := norm_add_le _ _ |
| _ = ((1+a)/2)*‖v‖+((1-a)/2)*‖w‖ := by |
| rw [norm_smul,norm_smul,Real.norm_of_nonneg hn1,Real.norm_of_nonneg hn2] |
| _ ≤ ((1+a)/2)*R+((1-a)/2)*R := add_le_add |
| (mul_le_mul_of_nonneg_left hv hn1) (mul_le_mul_of_nonneg_left hw hn2) |
| _ = R := by ring |
| apply max_le |
| · have he : u+a • v=((1+a)/2) • (u+v)+((1-a)/2) • (u-v) := by module |
| rw [he] |
| exact hn _ _ hp hm |
| · have he : u-a • v=((1+a)/2) • (u-v)+((1-a)/2) • (u+v) := by module |
| rw [he] |
| exact hn _ _ hm hp |
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| / |
| def LocalQuotientGain (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] |
| (t g : ℝ) : Prop := |
| ∀ u : X,‖u‖=1 → ∃ F : Submodule ℝ X,IsClosed (F : Set X) ∧ |
| FiniteDimensional ℝ (X ⧸ F) ∧ |
| ∀ y : X,y ∈ F → ‖y‖=1 → 1+g ≤ max ‖u+t • y‖ ‖u-t • y‖ |
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| lemma LocalQuotientGain.large_step {t g : ℝ} (ht : 0 < t) |
| (h : LocalQuotientGain X t g) : QuotientGain X t g := by |
| intro u hu |
| obtain ⟨F,hclosed,hfinite,hF⟩ := h u hu |
| refine ⟨F,hclosed,hfinite,?_⟩ |
| intro v hv htv |
| have hv0 : 0 < ‖v‖ := ht.trans_le htv |
| have hunit : ‖‖v‖⁻¹ • v‖=1 := by |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hv0.le),inv_mul_cancel₀ hv0.ne'] |
| have hl := hF (‖v‖⁻¹ • v) (F.smul_mem _ hv) hunit |
| have hle := lens_shrink (show 0 ≤ t/‖v‖ by positivity) |
| ((div_le_one₀ hv0).mpr htv) u v |
| (le_max_left ‖u+v‖ ‖u-v‖) (le_max_right ‖u+v‖ ‖u-v‖) |
| simp only [smul_smul,← div_eq_mul_inv] at hl |
| exact hl.trans hle |
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| / |
| retained, not replaced by a coordinate-null convention. -/ |
| def Cof (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] := |
| {F : Submodule ℝ X // IsClosed (F : Set X) ∧ FiniteDimensional ℝ (X ⧸ F)} |
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| def maximumModulus (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] (t : ℝ) : EReal := |
| ⨅ u : {u : X // ‖u‖=1}, ⨆ F : Cof X, |
| ⨅ y : {y : X // y ∈ F.val ∧ ‖y‖=1}, |
| ((max ‖u.val+t • y.val‖ ‖u.val-t • y.val‖-1 : ℝ) : EReal) |
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| def AMUC (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop := |
| ∀ t : ℝ,0 < t → t < 1 → 0 < maximumModulus X t |
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| lemma positive_modulus_gain {t : ℝ} (h : 0 < maximumModulus X t) : |
| ∃ g : ℝ,0 < g ∧ LocalQuotientGain X t g := by |
| obtain ⟨g,hg,hgδ⟩ := EReal.exists_between_coe_real h |
| refine ⟨g,by exact_mod_cast hg,?_⟩ |
| intro u hu |
| let M : {w : X // ‖w‖=1} → EReal := fun w => |
| ⨆ F : Cof X, ⨅ y : {y : X // y ∈ F.val ∧ ‖y‖=1}, |
| ((max ‖w.val+t • y.val‖ ‖w.val-t • y.val‖-1 : ℝ) : EReal) |
| have hmu : maximumModulus X t ≤ M ⟨u,hu⟩ := |
| iInf_le M (⟨u,hu⟩ : {w : X // ‖w‖=1}) |
| have hgu := hgδ.trans_le hmu |
| change (g : EReal) < ⨆ F : Cof X, |
| ⨅ y : {y : X // y ∈ F.val ∧ ‖y‖=1}, |
| ((max ‖u+t • y.val‖ ‖u-t • y.val‖-1 : ℝ) : EReal) at hgu |
| obtain ⟨F,hF⟩ := lt_iSup_iff.mp hgu |
| refine ⟨F.val,F.property.1,F.property.2,?_⟩ |
| intro y hyF hy1 |
| have hvalue : (⨅ z : {z : X // z ∈ F.val ∧ ‖z‖=1}, |
| ((max ‖u+t • z.val‖ ‖u-t • z.val‖-1 : ℝ) : EReal)) ≤ |
| ((max ‖u+t • y‖ ‖u-t • y‖-1 : ℝ) : EReal) := |
| iInf_le (fun z : {z : X // z ∈ F.val ∧ ‖z‖=1} => |
| ((max ‖u+t • z.val‖ ‖u-t • z.val‖-1 : ℝ) : EReal)) |
| (⟨y,hyF,hy1⟩ : {z : X // z ∈ F.val ∧ ‖z‖=1}) |
| have hb := hF.trans_le hvalue |
| have hh : g < max ‖u+t • y‖ ‖u-t • y‖-1 := by exact_mod_cast hb |
| linarith |
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| end Midpoint |
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| namespace Diamond |
| variable {X Y : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] |
| [NormedAddCommGroup Y] [NormedSpace ℝ Y] |
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| / |
| diamonds. Finite dimensional spaces are automatically covered as well. -/ |
| theorem amuc_obstruction (hX : Midpoint.AMUC X) : ¬ UniformEmbeddings X := by |
| rintro ⟨D,h⟩ |
| obtain ⟨f₀,s₀,h₀⟩ := h 0 |
| have hD : 0 < D := lt_of_lt_of_le (by norm_num) h₀.1 |
| let t := 1/(2*D) |
| have ht : 0 < t := by dsimp [t]; positivity |
| have ht1 : t < 1 := (div_lt_one (by positivity : 0 < 2*D)).mpr (by linarith [h₀.1]) |
| obtain ⟨g,hg,hgX⟩ := Midpoint.positive_modulus_gain (hX t ht ht1) |
| obtain ⟨k,hk'⟩ := exists_nat_gt (D/g) |
| have hkg : D < (k:ℝ)*g := (div_lt_iff₀ hg).mp hk' |
| have hBern := one_add_mul_le_pow (by linarith : (-2:ℝ) ≤ g) k |
| have hk : D < (1+g)^k := by linarith |
| obtain ⟨f,s,hf⟩ := h k |
| have hh := quotient_full_gain ht hg (le_refl t) (hgX.large_step ht) k f hf |
| have hp : 0 < (1+g)^k := by positivity |
| have hpow : (1+g)^k ≤ D := by simpa only [one_mul] using (le_div_iff₀ hp).mp hh |
| exact (not_lt_of_ge hpow) hk |
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| / |
| equivalence is simply the identification of the original space with its |
| equivalent-norm copy; there is no change of graph or choice of scales. -/ |
| lemma equivalent_distortion (e : X ≃ₗ[ℝ] Y) {a b : ℝ} (ha : 0 < a) (hb : 0 < b) |
| (he : ∀ x : X,a*‖x‖ ≤ ‖e x‖ ∧ ‖e x‖ ≤ b*‖x‖) |
| {k : ℕ} (f : Vertex k → X) {D s : ℝ} (hf : HasDistortion k f D s) : |
| HasDistortion k (fun v => e (f v)) ((b/a)*D) (a*s) := by |
| have hab : a ≤ b := by |
| have hl := slope_lower f (fun u v => (hf.2.2 u v).1) (baseEdge k) |
| have hn : 0 < ‖slope f (baseEdge k)‖ := hf.2.1.trans_le hl |
| have h := (he (slope f (baseEdge k))).1.trans (he (slope f (baseEdge k))).2 |
| exact (mul_le_mul_iff_left₀ hn).mp (by simpa only [mul_comm] using h) |
| have hba : 1 ≤ b/a := (one_le_div ha).mpr hab |
| refine ⟨?_,mul_pos ha hf.2.1,?_⟩ |
| · nlinarith [hf.1] |
| · intro u v |
| rw [← map_sub] |
| constructor |
| · have hh := mul_le_mul_of_nonneg_left (hf.2.2 u v).1 ha.le |
| have hh' : (a*s)*distance k u v ≤ a*‖f u-f v‖ := by simpa only [mul_assoc] using hh |
| exact hh'.trans (he (f u-f v)).1 |
| · have hh := ((he (f u-f v)).2).trans |
| (mul_le_mul_of_nonneg_left (hf.2.2 u v).2 hb.le) |
| have hid : b*(D*s*distance k u v)=((b/a)*D)*(a*s)*distance k u v := by |
| field_simp |
| rwa [hid] at hh |
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| theorem equivalent_amuc [CompleteSpace X] [CompleteSpace Y] |
| (e : X ≃ₗ[ℝ] Y) {a b : ℝ} (ha : 0 < a) (hb : 0 < b) |
| (he : ∀ x : X,a*‖x‖ ≤ ‖e x‖ ∧ ‖e x‖ ≤ b*‖x‖) |
| (hY : Midpoint.AMUC Y) : ¬ UniformEmbeddings X := by |
| rintro ⟨D,h⟩ |
| apply amuc_obstruction hY |
| refine ⟨(b/a)*D,?_⟩ |
| intro k |
| obtain ⟨f,s,hf⟩ := h k |
| exact ⟨fun v => e (f v),a*s,equivalent_distortion e ha hb he f hf⟩ |
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| end Diamond |
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| end DiamondDistortion |
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| end |
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| end OAI |
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