| import OAI.Geometry.ProjectionVolume.ProductVolume |
| import OAI.Geometry.ProjectionVolume.SimplexProjection |
| import Mathlib.Analysis.InnerProductSpace.NormDet |
| import Mathlib.LinearAlgebra.Matrix.SchurComplement |
| |
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
| open scoped BigOperators RealInnerProductSpace ENNReal |
|
|
| namespace Paper092 |
|
|
| def facetEmbedding (n : ℕ) (i : Fin (n + 1)) (c : ℝ) : |
| Euclidean n →ₗ[ℝ] Euclidean (n + 1) where |
| toFun x := WithLp.toLp 2 (i.insertNth (c * ∑ j, x j) x) |
| map_add' x y := by |
| ext j |
| induction j using i.succAboveCases <;> |
| simp [PiLp.add_apply, Finset.sum_add_distrib, mul_add] |
| map_smul' a x := by |
| ext j |
| induction j using i.succAboveCases < |
| simp [PiLp.smul_apply, smul_eq_mul, Finset.mul_sum, mul_left_comm] |
| |
| theorem facetEmbedding_apply_same (n : ℕ) (i : Fin (n + 1)) (c : ℝ) (x : Euclidean n) : |
| facetEmbedding n i c x i = c * ∑ j, x j := by |
| simp [facetEmbedding] |
|
|
| theorem facetEmbedding_apply_succAbove (n : ℕ) (i : Fin (n + 1)) (c : ℝ) |
| (x : Euclidean n) (j : Fin n) : |
| facetEmbedding n i c x (i.succAbove j) = x j := by |
| simp [facetEmbedding] |
|
|
| theorem facetEmbedding_inner (n : ℕ) (i : Fin (n + 1)) (c : ℝ) (x y : Euclidean n) : |
| ⟪facetEmbedding n i c x, facetEmbedding n i c y⟫ = |
| ⟪x, y⟫ + c ^ 2 * (∑ j, x j) * (∑ j, y j) := by |
| rw [PiLp.inner_apply, Fin.sum_univ_succAbove _ i] |
| simp only [facetEmbedding_apply_same, facetEmbedding_apply_succAbove, Real.inner_apply, |
| PiLp.inner_apply] |
| ring |
|
|
| theorem facetEmbedding_normDet_sq (n : ℕ) (i : Fin (n + 1)) (c : ℝ) : |
| (facetEmbedding n i c).normDet ^ 2 = 1 + (n : ℝ) * c ^ 2 := by |
| have hsq := (facetEmbedding n i c).normDet_sq_eq_det_gram (EuclideanSpace.basisFun (Fin n) ℝ) |
| simp only [RCLike.ofReal_real_eq_id, id_eq] at hsq |
| have hgram : Matrix.gram ℝ |
| (fun j => facetEmbedding n i c (EuclideanSpace.basisFun (Fin n) ℝ j)) = |
| 1 + Matrix.replicateCol (Fin 1) (fun _ : Fin n => c) * |
| Matrix.replicateRow (Fin 1) (fun _ : Fin n => c) := by |
| ext j k |
| simp only [Matrix.gram_apply, facetEmbedding_inner, |
| (EuclideanSpace.basisFun (Fin n) ℝ).inner_eq_ite] |
| simp [EuclideanSpace.basisFun_apply, PiLp.single_apply, Matrix.one_apply, |
| Matrix.mul_apply, pow_two] |
| rw [hgram] at hsq |
| have hd := Matrix.det_one_add_replicateCol_mul_replicateRow (ι := Fin 1) |
| (fun _ : Fin n => c) (fun _ : Fin n => c) |
| simpa [dotProduct, pow_two, mul_assoc] using hsq.trans hd |
|
|
| theorem facetEmbedding_normDet_zero (n : ℕ) (i : Fin (n + 1)) : |
| (facetEmbedding n i 0).normDet = 1 := by |
| have h := facetEmbedding_normDet_sq n i 0 |
| have hp := (facetEmbedding n i 0).normDet_nonneg |
| nlinarith |
|
|
| theorem facetEmbedding_normDet_neg_one (n : ℕ) (i : Fin (n + 1)) : |
| (facetEmbedding n i (-1)).normDet = Real.sqrt (n + 1) := by |
| have h := facetEmbedding_normDet_sq n i (-1) |
| have hp := (facetEmbedding n i (-1)).normDet_nonneg |
| have hs := Real.sq_sqrt (by positivity : (0 : ℝ) ≤ n + 1) |
| have hsp := Real.sqrt_nonneg ((n : ℝ) + 1) |
| nlinarith |
|
|
| theorem facetEmbedding_zero_image (n : ℕ) (i : Fin (n + 1)) : |
| facetEmbedding n i 0 '' standardSimplex n = simplexFacet (n + 1) (some i) := by |
| ext y |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| change facetEmbedding n i 0 x ∈ standardSimplex (n + 1) ∧ |
| facetEmbedding n i 0 x i = 0 |
| rw [mem_standardSimplex_iff] at hx ⊢ |
| simp [facetEmbedding, Fin.forall_iff_succAbove i, hx] |
| · rintro ⟨hy, hi⟩ |
| change y i = 0 at hi |
| refine ⟨WithLp.toLp 2 (i.removeNth y), ?_, ?_⟩ |
| · rw [mem_standardSimplex_iff] at hy ⊢ |
| refine ⟨fun j => hy.1 (i.succAbove j), ?_⟩ |
| have hsum := Fin.add_sum_removeNth i (fun j => y j) |
| simpa [hi, Fin.removeNth] using hsum.trans_le hy.2 |
| · ext j |
| induction j using i.succAboveCases <;> simp [facetEmbedding, hi] |
| |
| theorem simplexFacet_coordinate_area (n : ℕ) (i : Fin (n + 1)) : |
| Measure.euclideanHausdorffMeasure n (simplexFacet (n + 1) (some i)) = |
| ENNReal.ofReal (1 / (Nat.factorial n : ℝ)) := by |
| rw [← facetEmbedding_zero_image n i] |
| have h := (facetEmbedding n i 0).euclideanHausdorffMeasure_image_eq_normDet_mul_volume |
| (standardSimplex n) |
| simpa only [finrank_euclideanSpace_fin, facetEmbedding_normDet_zero, ENNReal.ofReal_one, |
| one_mul, standardSimplex_volume] using h |
| |
| def diagonalFacetChart (n : ℕ) (i : Fin (n + 1)) (x : Euclidean n) : Euclidean (n + 1) := |
| WithLp.toLp 2 (i.insertNth (1 - ∑ j, x j) x) |
| |
| theorem diagonalFacetChart_eq (n : ℕ) (i : Fin (n + 1)) (x : Euclidean n) : |
| diagonalFacetChart n i x = facetEmbedding n i (-1) x + EuclideanSpace.single i 1 := by |
| ext j |
| induction j using i.succAboveCases <;> |
| simp [diagonalFacetChart, facetEmbedding, sub_eq_add_neg, add_comm] |
| |
| theorem diagonalFacetChart_image (n : ℕ) (i : Fin (n + 1)) : |
| diagonalFacetChart n i '' standardSimplex n = simplexFacet (n + 1) none := by |
| ext y |
| constructor |
| · rintro ⟨x, hx, rfl⟩ |
| rw [mem_standardSimplex_iff] at hx |
| change diagonalFacetChart n i x ∈ standardSimplex (n + 1) ∧ |
| 1 - ∑ j, diagonalFacetChart n i x j = 0 |
| rw [mem_standardSimplex_iff] |
| simp [diagonalFacetChart, Fin.forall_iff_succAbove i] |
| exact ⟨hx.2, hx.1⟩ |
| · rintro ⟨hy, hi⟩ |
| change 1 - ∑ j, y j = 0 at hi |
| have htotal : ∑ j, y j = 1 := by linarith |
| rw [mem_standardSimplex_iff] at hy |
| have hsum := Fin.add_sum_removeNth i (fun j => y j) |
| refine ⟨WithLp.toLp 2 (i.removeNth y), ?_, ?_⟩ |
| · rw [mem_standardSimplex_iff] |
| refine ⟨fun j => hy.1 (i.succAbove j), ?_⟩ |
| change ∑ j, i.removeNth (fun k => y k) j ≤ 1 |
| linarith [hy.1 i] |
| · ext j |
| induction j using i.succAboveCases |
| · simp only [diagonalFacetChart, PiLp.toLp_apply, Fin.insertNth_apply_same] |
| change 1 - ∑ j, i.removeNth (fun k => y k) j = y i |
| linarith |
| · simp [diagonalFacetChart] |
| |
| theorem simplexFacet_diagonal_area (n : ℕ) : |
| Measure.euclideanHausdorffMeasure n (simplexFacet (n + 1) none) = |
| ENNReal.ofReal (Real.sqrt (n + 1) / (Nat.factorial n : ℝ)) := by |
| let i : Fin (n + 1) := Fin.last n |
| rw [← diagonalFacetChart_image n i] |
| have hset : diagonalFacetChart n i '' standardSimplex n = |
| (IsometryEquiv.vaddConst (EuclideanSpace.single i (1 : ℝ))) '' |
| (facetEmbedding n i (-1) '' standardSimplex n) := by |
| rw [image_image] |
| congr 1 |
| funext x |
| exact diagonalFacetChart_eq n i x |
| rw [hset, (IsometryEquiv.vaddConst (EuclideanSpace.single i (1 : ℝ))).isometry.euclideanHausdorffMeasure_image] |
| have h := (facetEmbedding n i (-1)).euclideanHausdorffMeasure_image_eq_normDet_mul_volume |
| (standardSimplex n) |
| simp only [finrank_euclideanSpace_fin, facetEmbedding_normDet_neg_one, |
| standardSimplex_volume] at h |
| rw [h, ← ENNReal.ofReal_mul (Real.sqrt_nonneg _)] |
| congr 1 |
| ring |
| |
| end Paper092 |
| |
| end |
| |
| end OAI |
| |
