| 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 | |