| import Mathlib | |
| import OAI.Probability.LatticeSurface.Convexity.BrunnMinkowski | |
| import OAI.Probability.LatticeSurface.Convexity.CubePrekopa | |
| import OAI.Probability.LatticeSurface.Convexity.QuantileTransport | |
| import OAI.Probability.LatticeSurface.Convexity.WeightedPrekopa | |
| import OAI.Probability.LatticeSurface.Dynamics.DisplacementInequality | |
| namespace OAI | |
| open Set MeasureTheory Filter | |
| open scoped Topology | |
| noncomputable section | |
| namespace FacetVolume | |
| end FacetVolume | |
| namespace ExtremalEdges | |
| open scoped ENNReal | |
| lemma finite_volume_brunn_minkowski {I : Type*} [Fintype I] | |
| {K L M : Set (I → ℝ)} {s t : ℝ} | |
| (hK : IsCompact K) (hL : IsCompact L) (hM : IsCompact M) | |
| (hKn : K.Nonempty) (hLn : L.Nonempty) (hs : 0 ≤ s) (ht : 0 ≤ t) (hst : s+t=1) | |
| (hcomb : ∀ x ∈ K, ∀ y ∈ L, s • x+t • y ∈ M) : | |
| ((volume K).toReal)^s*((volume L).toReal)^t ≤ (volume M).toReal := by | |
| let e := MeasurableEquiv.piCongrLeft (fun _ : Fin (Fintype.card I) => ℝ) (Fintype.equivFin I) | |
| have hec : Continuous e := (Homeomorph.piCongrLeft (Fintype.equivFin I)).continuous | |
| have hemp := (volume_measurePreserving_piCongrLeft | |
| (fun _ : Fin (Fintype.card I) => ℝ) (Fintype.equivFin I)).symm e | |
| have hv (S : Set (I → ℝ)) (hS : IsCompact S) : volume (e '' S) = volume S := by | |
| rw [e.image_eq_preimage_symm] | |
| exact hemp.measure_preimage hS.measurableSet.nullMeasurableSet | |
| have hm := FacetVolume.compact_volume_brunn_minkowski | |
| (hK.image hec) (hL.image hec) (hM.image hec) (hKn.image _) (hLn.image _) hs ht hst | |
| (fun x hx y hy => by | |
| rcases hx with ⟨a,ha,rfl⟩ | |
| rcases hy with ⟨b,hb,rfl⟩ | |
| refine ⟨s • a+t • b,hcomb a ha b hb,?_⟩ | |
| funext i | |
| simp [e, MeasurableEquiv.piCongrLeft, Equiv.piCongrLeft_apply]) | |
| simpa only [hv K hK, hv L hL, hv M hM] using hm | |
| def productSlice {I J : Type*} (K : Set ((I → ℝ) × (J → ℝ))) (a : J → ℝ) : | |
| Set (I → ℝ) := {h | (h,a) ∈ K} | |
| lemma compact_productSlice {I J : Type*} [Fintype I] [Fintype J] | |
| {K : Set ((I → ℝ) × (J → ℝ))} (hK : IsCompact K) (a : J → ℝ) : | |
| IsCompact (productSlice K a) := by | |
| have he : Prod.fst '' (K ∩ {p | p.2 = a}) = productSlice K a := by | |
| ext h | |
| constructor | |
| · rintro ⟨⟨h,b⟩,⟨hh,hb⟩,rfl⟩ | |
| dsimp at hb | |
| subst b | |
| exact hh | |
| · intro hh; exact ⟨(h,a),⟨hh,rfl⟩,rfl⟩ | |
| rw [← he] | |
| exact (hK.inter_right (isClosed_eq continuous_snd continuous_const)).image continuous_fst | |
| lemma volume_productSlice_neg {I J : Type*} [Fintype I] [Fintype J] | |
| {K : Set ((I → ℝ) × (J → ℝ))} (hK : IsCompact K) | |
| (hn : ∀ p ∈ K, -p ∈ K) (a : J → ℝ) : | |
| volume (productSlice K (-a)) = volume (productSlice K a) := by | |
| have he : productSlice K (-a) = Neg.neg ⁻¹' productSlice K a := by | |
| ext h | |
| change (h,-a) ∈ K ↔ (-h,a) ∈ K | |
| constructor | |
| · intro hh; simpa using hn _ hh | |
| · intro hh; simpa using hn _ hh | |
| rw [he] | |
| exact (Measure.measurePreserving_neg volume).measure_preimage | |
| (compact_productSlice hK a).measurableSet.nullMeasurableSet | |
| theorem central_productSlice_max {I J : Type*} [Fintype I] [Fintype J] | |
| {K : Set ((I → ℝ) × (J → ℝ))} (hK : IsCompact K) (hc : Convex ℝ K) | |
| (hn : ∀ p ∈ K, -p ∈ K) (a : J → ℝ) : | |
| volume (productSlice K a) ≤ volume (productSlice K 0) := by | |
| apply (ENNReal.toReal_le_toReal (compact_productSlice hK a).measure_ne_top | |
| (compact_productSlice hK 0).measure_ne_top).mp | |
| by_cases hz : (volume (productSlice K a)).toReal = 0 | |
| · rw [hz]; positivity | |
| have hp : 0 < (volume (productSlice K a)).toReal := lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm hz) | |
| have hsn : (productSlice K a).Nonempty := by | |
| by_contra hh | |
| simp [Set.not_nonempty_iff_eq_empty.mp hh] at hp | |
| have hnn : (productSlice K (-a)).Nonempty := by | |
| obtain ⟨x,hx⟩ := hsn | |
| exact ⟨-x, by simpa only [productSlice, mem_ofPred_eq, Prod.neg_mk] using hn (x,a) hx⟩ | |
| have hb := finite_volume_brunn_minkowski (compact_productSlice hK a) | |
| (compact_productSlice hK (-a)) (compact_productSlice hK 0) hsn hnn | |
| (s := 1/2) (t := 1/2) (by norm_num) (by norm_num) (by norm_num) | |
| (fun x hx y hy => by | |
| have hh := hc hx hy (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (0:ℝ) ≤ 1/2) | |
| (by norm_num : (1/2:ℝ)+1/2=1) | |
| have he : (1/2:ℝ) • (x,a)+(1/2:ℝ) • (y,-a) = ((1/2:ℝ) • x+(1/2:ℝ) • y,0) := by | |
| ext i <;> simp | |
| change ((1/2:ℝ) • x+(1/2:ℝ) • y, 0) ∈ K | |
| rw [← he] | |
| exact hh) | |
| rw [volume_productSlice_neg hK hn, ← Real.rpow_add hp] at hb | |
| norm_num at hb | |
| exact hb | |
| theorem product_small_set_bound {I J : Type*} [Fintype I] [Fintype J] | |
| {K : Set ((I → ℝ) × (J → ℝ))} (hK : IsCompact K) (hc : Convex ℝ K) | |
| (hn : ∀ p ∈ K, -p ∈ K) (S : Set (J → ℝ)) (hS : MeasurableSet S) : | |
| volume (K ∩ Prod.snd ⁻¹' S) ≤ volume S * volume (productSlice K 0) := by | |
| change ((volume : Measure (I → ℝ)).prod (volume : Measure (J → ℝ))) _ ≤ _ | |
| rw [Measure.prod_apply_symm (hK.measurableSet.inter (hS.preimage measurable_snd))] | |
| calc | |
| (∫⁻ a, volume ((fun h : I → ℝ => (h,a)) ⁻¹' (K ∩ Prod.snd ⁻¹' S))) ≤ | |
| ∫⁻ a, S.indicator (fun _ => volume (productSlice K 0)) a := by | |
| apply lintegral_mono | |
| intro a | |
| by_cases ha : a ∈ S | |
| · rw [Set.indicator_of_mem ha] | |
| have he : ((fun h : I → ℝ => (h,a)) ⁻¹' (K ∩ Prod.snd ⁻¹' S)) = productSlice K a := by | |
| ext h; simp [productSlice, ha] | |
| simpa only [he] using central_productSlice_max hK hc hn a | |
| · rw [Set.indicator_of_notMem ha] | |
| have he : ((fun h : I → ℝ => (h,a)) ⁻¹' (K ∩ Prod.snd ⁻¹' S)) = ∅ := by | |
| ext h; simp [ha] | |
| simp only [he, measure_empty, le_refl] | |
| _ = _ := by rw [lintegral_indicator hS, lintegral_const, Measure.restrict_apply_univ, mul_comm] | |
| end ExtremalEdges | |
| end | |
| end OAI | |