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cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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