| import OAI.Geometry.ProjectionVolume.ProductPolytope | |
| import Mathlib.Analysis.Convex.Exposed | |
| import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| namespace Paper092 | |
| def exposedFace {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) : Set (Euclidean n) := | |
| (innerSL ℝ u).toExposed K | |
| @[simp] theorem exposedFace_zero {n : ℕ} (K : Set (Euclidean n)) : exposedFace K 0 = K := by | |
| ext x | |
| simp [exposedFace, ContinuousLinearMap.toExposed] | |
| theorem exposedFace_nonempty {n : ℕ} (K : Set (Euclidean n)) (hK : IsCompact K) | |
| (hne : K.Nonempty) (u : Euclidean n) : (exposedFace K u).Nonempty := by | |
| obtain ⟨x, hx, hmax⟩ := hK.exists_isMaxOn hne (innerSL ℝ u).continuous.continuousOn | |
| exact ⟨x, hx, hmax⟩ | |
| theorem exposedFace_product {r s : ℕ} (A : Set (Euclidean r)) (B : Set (Euclidean s)) | |
| (w : Euclidean (r + s)) : | |
| exposedFace (cartesianBody A B) w = | |
| cartesianBody (exposedFace A (splitEuclideanProduct r s w).1) | |
| (exposedFace B (splitEuclideanProduct r s w).2) := by | |
| ext x | |
| change (x ∈ cartesianBody A B ∧ ∀ y ∈ cartesianBody A B, ⟪w, y⟫ ≤ ⟪w, x⟫) ↔ | |
| (((splitEuclideanProduct r s x).1 ∈ A ∧ | |
| ∀ a ∈ A, ⟪(splitEuclideanProduct r s w).1, a⟫ ≤ | |
| ⟪(splitEuclideanProduct r s w).1, (splitEuclideanProduct r s x).1⟫) ∧ | |
| ((splitEuclideanProduct r s x).2 ∈ B ∧ | |
| ∀ b ∈ B, ⟪(splitEuclideanProduct r s w).2, b⟫ ≤ | |
| ⟪(splitEuclideanProduct r s w).2, (splitEuclideanProduct r s x).2⟫)) | |
| constructor | |
| · rintro ⟨⟨hA, hB⟩, hmax⟩ | |
| refine ⟨⟨hA, ?_⟩, ⟨hB, ?_⟩⟩ | |
| · intro a ha | |
| have hm : (splitEuclideanProduct r s).symm (a, (splitEuclideanProduct r s x).2) ∈ | |
| cartesianBody A B := by | |
| rw [cartesianBody_eq_preimage, Set.mem_preimage, | |
| ContinuousLinearEquiv.apply_symm_apply] | |
| exact ⟨ha, hB⟩ | |
| have h := hmax _ hm | |
| rw [splitEuclideanProduct_inner r s, splitEuclideanProduct_inner r s, | |
| ContinuousLinearEquiv.apply_symm_apply] at h | |
| exact (add_le_add_iff_right _).mp h | |
| · intro b hb | |
| have hm : (splitEuclideanProduct r s).symm ((splitEuclideanProduct r s x).1, b) ∈ | |
| cartesianBody A B := by | |
| rw [cartesianBody_eq_preimage, Set.mem_preimage, | |
| ContinuousLinearEquiv.apply_symm_apply] | |
| exact ⟨hA, hb⟩ | |
| have h := hmax _ hm | |
| rw [splitEuclideanProduct_inner r s, splitEuclideanProduct_inner r s, | |
| ContinuousLinearEquiv.apply_symm_apply] at h | |
| exact (add_le_add_iff_left _).mp h | |
| · rintro ⟨⟨hA, hmaxA⟩, ⟨hB, hmaxB⟩⟩ | |
| refine ⟨⟨hA, hB⟩, ?_⟩ | |
| intro y hy | |
| rw [splitEuclideanProduct_inner r s, splitEuclideanProduct_inner r s] | |
| exact add_le_add (hmaxA _ hy.1) (hmaxB _ hy.2) | |
| theorem cartesianBody_eq_image {r s : ℕ} (A : Set (Euclidean r)) (B : Set (Euclidean s)) : | |
| cartesianBody A B = (splitEuclideanProduct r s).symm '' (A ×ˢ B) := by | |
| ext x | |
| constructor | |
| · intro hx | |
| exact ⟨splitEuclideanProduct r s x, hx, (splitEuclideanProduct r s).symm_apply_apply x⟩ | |
| · rintro ⟨y, hy, rfl⟩ | |
| change splitEuclideanProduct r s ((splitEuclideanProduct r s).symm y) ∈ A ×ˢ B | |
| simpa using hy | |
| theorem submodule_prod_finrank {r s : ℕ} (P : Submodule ℝ (Euclidean r)) | |
| (Q : Submodule ℝ (Euclidean s)) : | |
| Module.finrank ℝ (P.prod Q) = Module.finrank ℝ P + Module.finrank ℝ Q := by | |
| let e : (P.prod Q) ≃ₗ[ℝ] P × Q := | |
| { toFun := fun x => (⟨x.1.1, x.2.1⟩, ⟨x.1.2, x.2.2⟩) | |
| invFun := fun x => ⟨(x.1, x.2), ⟨x.1.2, x.2.2⟩⟩ | |
| left_inv := fun _ => rfl | |
| right_inv := fun _ => rfl | |
| map_add' := fun _ _ => rfl | |
| map_smul' := fun _ _ => rfl } | |
| rw [e.finrank_eq, Module.finrank_prod] | |
| theorem cartesianBody_affine_dimension {r s : ℕ} (A : Set (Euclidean r)) (B : Set (Euclidean s)) | |
| (hA : A.Nonempty) (hB : B.Nonempty) : | |
| Module.finrank ℝ (vectorSpan ℝ (cartesianBody A B)) = | |
| Module.finrank ℝ (vectorSpan ℝ A) + Module.finrank ℝ (vectorSpan ℝ B) := by | |
| let e := (splitEuclideanProduct r s).symm.toLinearEquiv | |
| rw [cartesianBody_eq_image] | |
| change Module.finrank ℝ (vectorSpan ℝ (e.toLinearMap.toAffineMap '' (A ×ˢ B))) = _ | |
| rw [← AffineMap.map_vectorSpan] | |
| change Module.finrank ℝ ((vectorSpan ℝ (A ×ˢ B)).map e.toLinearMap) = _ | |
| rw [e.finrank_map_eq, vectorSpan_prod_eq hA hB, submodule_prod_finrank] | |
| end Paper092 | |
| end | |
| end OAI | |