| import OAI.Geometry.ProjectionVolume.PolytopeFaceGeometry |
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| namespace OAI |
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| universe uι |
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| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
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| noncomputable section |
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| namespace Paper092.HPolytope |
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| variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) |
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| theorem affineSpan_face_of_essential (i : ι) |
| (hessential : ¬ ∀ x : Euclidean d, |
| (∀ j, j ≠ i → ⟪P.normal j, x⟫ ≤ P.offset j) → ⟪P.normal i, x⟫ ≤ P.offset i) : |
| (affineSpan ℝ (P.face i) : Set (Euclidean d)) = |
| {y | ⟪P.normal i, y⟫ = P.offset i} := by |
| classical |
| let Q : AffineSubspace ℝ (Euclidean d) := |
| (affineSpan ℝ ({P.offset i} : Set ℝ)).comap (innerₛₗ ℝ (P.normal i)).toAffineMap |
| have hQ (y : Euclidean d) : y ∈ Q ↔ ⟪P.normal i, y⟫ = P.offset i := by |
| simp [Q] |
| have hle : affineSpan ℝ (P.face i) ≤ Q := by |
| apply affineSpan_le.mpr |
| intro y hy |
| exact (hQ y).mpr hy.2 |
| ext y |
| constructor |
| · exact fun hy => (hQ y).mp (hle hy) |
| · intro hy |
| change ⟪P.normal i, y⟫ = P.offset i at hy |
| change y ∈ affineSpan ℝ (P.face i) |
| obtain ⟨x, hx, hxstrict⟩ := P.exists_strict_face_point_of_essential i hessential |
| have hxspan : x ∈ affineSpan ℝ (P.face i) := subset_affineSpan ℝ _ hx |
| by_cases hxy : y = x |
| · simpa only [hxy] using hxspan |
| let S : Set (Euclidean d) := ⋂ j : {j : ι // j ≠ i}, |
| {z | ⟪P.normal j.val, z⟫ < P.offset j.val} |
| have hopen : IsOpen S := isOpen_iInter_of_finite fun j => |
| isOpen_lt (innerSL ℝ (P.normal j.val)).continuous continuous_const |
| have hxS : x ∈ S := mem_iInter.mpr (fun j => hxstrict j.val j.property) |
| obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp hopen x hxS |
| have hnorm : 0 < ‖y - x‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hxy) |
| let t : ℝ := ε / (2 * ‖y - x‖) |
| have ht : 0 < t := div_pos hε (mul_pos (by norm_num) hnorm) |
| let z := x + t • (y - x) |
| have hzball : z ∈ Metric.ball x ε := by |
| rw [Metric.mem_ball, dist_eq_norm] |
| change ‖x + t • (y - x) - x‖ < ε |
| rw [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_pos ht] |
| dsimp [t] |
| have heq : ε / (2 * ‖y - x‖) * ‖y - x‖ = ε / 2 := by field_simp |
| rw [heq] |
| linarith |
| have hzS := hball hzball |
| have hzi : ⟪P.normal i, z⟫ = P.offset i := by |
| dsimp [z] |
| simp only [inner_add_right, real_inner_smul_right, inner_sub_right, hy, hx.2, |
| sub_self, mul_zero, add_zero] |
| have hzface : z ∈ P.face i := by |
| refine ⟨?_, hzi⟩ |
| intro j |
| by_cases hji : j = i |
| · simpa only [hji, hzi] using le_refl (P.offset i) |
| · exact (mem_iInter.mp hzS ⟨j, hji⟩).le |
| have hzspan := subset_affineSpan ℝ (P.face i) hzface |
| have hmem := (affineSpan ℝ (P.face i)).smul_vsub_vadd_mem t⁻¹ hzspan hxspan hxspan |
| have heq : t⁻¹ • (z - x) + x = y := by |
| dsimp [z] |
| rw [add_sub_cancel_left, smul_smul, inv_mul_cancel₀ ht.ne', one_smul, sub_add_cancel] |
| simpa only [vsub_eq_sub, vadd_eq_add, heq] using hmem |
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| end Paper092.HPolytope |
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| end |
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| end OAI |
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