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2fca478 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Geometry.ProjectionBody.GeometryDefinitions
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
import Mathlib.Analysis.InnerProductSpace.PiL2
namespace OAI
open Set
namespace ProjectionCounterexample
noncomputable section
theorem simplex_convex (d : ℕ) : Convex ℝ (simplex d) :=
convex_convexHull ℝ _
theorem simplex_compact (d : ℕ) : IsCompact (simplex d) :=
((finite_range _).insert 0).isCompact_convexHull ℝ
theorem simplex_interior_nonempty (d : ℕ) : (interior (simplex d)).Nonempty := by
rw [(simplex_convex d).interior_nonempty_iff_affineSpan_eq_top]
rw [simplex, affineSpan_convexHull]
apply SetLike.coe_injective
rw [affineSpan_insert_zero]
have hb : (EuclideanSpace.basisFun (Fin d) ℝ : Fin d → E d) =
(fun i : Fin d => EuclideanSpace.single i (1 : ℝ)) := by
funext i
simp only [EuclideanSpace.basisFun_apply]
have hspan : Submodule.span ℝ (range fun i : Fin d => EuclideanSpace.single i (1 : ℝ)) = ⊤ := by
simpa only [OrthonormalBasis.coe_toBasis, hb] using
(EuclideanSpace.basisFun (Fin d) ℝ).toBasis.span_eq
exact congrArg (fun S : Submodule ℝ (E d) => (S : Set (E d))) hspan
theorem simplex_eq_coordinates (d : ℕ) :
simplex d = {x | (∀ i, 0 ≤ x i) ∧ ∑ i, x i ≤ 1} := by
classical
apply Subset.antisymm
· apply convexHull_min
· rintro x (rfl | ⟨i, rfl⟩)
· simp
· constructor
· intro j
simp only [PiLp.single_apply]
split_ifs
· exact zero_le_one
· exact le_rfl
· simp [PiLp.single_apply]
· intro x hx y hy a b ha hb hab
constructor
· intro i
exact add_nonneg (mul_nonneg ha (hx.1 i)) (mul_nonneg hb (hy.1 i))
· change ∑ i, (a * x i + b * y i) ≤ 1
rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum]
calc
a * ∑ i, x i + b * ∑ i, y i ≤ a * 1 + b * 1 :=
add_le_add (mul_le_mul_of_nonneg_left hx.2 ha)
(mul_le_mul_of_nonneg_left hy.2 hb)
_ = 1 := by simpa using hab
· intro x hx
let w : Option (Fin d) → ℝ := fun o =>
match o with
| none => 1 - ∑ i, x i
| some i => x i
let z : Option (Fin d) → E d := fun o =>
match o with
| none => 0
| some i => EuclideanSpace.single i 1
apply mem_convexHull_of_exists_fintype w z
· intro o
cases o with
| none => exact sub_nonneg.mpr hx.2
| some i => exact hx.1 i
· simp [w, Fintype.sum_option]
· intro o
cases o with
| none => exact mem_insert 0 _
| some i => exact mem_insert_of_mem 0 (mem_range_self i)
· simpa [w, z, Fintype.sum_option, EuclideanSpace.basisFun_apply] using
(EuclideanSpace.basisFun (Fin d) ℝ).sum_repr x
def blockEquiv : E 20 ≃L[ℝ] E 10 × E 10 :=
EuclideanSpace.finAddEquivProd (𝕜 := ℝ) (n := 10) (m := 10)
theorem blockEquiv_apply (x : E 20) :
blockEquiv x =
(WithLp.toLp 2 fun i : Fin 10 => x (Fin.castAdd 10 i),
WithLp.toLp 2 fun i : Fin 10 => x (Fin.natAdd 10 i)) := by
rfl
theorem productBody_eq_preimage :
productBody = blockEquiv ⁻¹' (simplex 10 ×ˢ simplex 10) := by
rfl
theorem productBody_compact : IsCompact productBody := by
rw [productBody_eq_preimage]
exact blockEquiv.toHomeomorph.isCompact_preimage.mpr
((simplex_compact 10).prod (simplex_compact 10))
theorem productBody_convex : Convex ℝ productBody := by
rw [productBody_eq_preimage]
exact ((simplex_convex 10).prod (simplex_convex 10)).linear_preimage
blockEquiv.toLinearMap
theorem productBody_interior_nonempty : (interior productBody).Nonempty := by
rw [productBody_eq_preimage]
change (interior (blockEquiv.toHomeomorph ⁻¹' (simplex 10 ×ˢ simplex 10))).Nonempty
rw [← blockEquiv.toHomeomorph.preimage_interior, interior_prod_eq]
obtain ⟨x, hx⟩ := simplex_interior_nonempty 10
exact ⟨blockEquiv.symm (x, x), by simpa using And.intro hx hx⟩
theorem productBody_convexBody :
IsCompact productBody ∧ Convex ℝ productBody ∧ (interior productBody).Nonempty :=
⟨productBody_compact, productBody_convex, productBody_interior_nonempty⟩
end
end ProjectionCounterexample
end OAI
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