| import OAI.Geometry.ProjectionVolume.PolytopeFront |
| import OAI.Geometry.ProjectionVolume.ExposedBounds |
| import Mathlib.Analysis.Convex.Intrinsic |
|
|
| namespace OAI |
|
|
| universe uι |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory |
| open scoped RealInnerProductSpace Pointwise |
|
|
| namespace Paper092 |
|
|
| theorem inner_eq_of_mem_intrinsicInterior {d : ℕ} {S : Set (Euclidean d)} |
| {x : Euclidean d} (hx : x ∈ intrinsicInterior ℝ S) (v : Euclidean d) |
| (hmax : ∀ z ∈ S, ⟪v, z⟫ ≤ ⟪v, x⟫) : |
| ∀ z ∈ S, ⟪v, z⟫ = ⟪v, x⟫ := by |
| obtain ⟨xQ, hxQ, rfl⟩ := mem_intrinsicInterior.mp hx |
| obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp hxQ) |
| intro z hz |
| by_cases hzx : z = xQ.val |
| · simp only [hzx] |
| let t : ℝ := ε / (2 * ‖xQ.val - z‖) |
| have hnorm : 0 < ‖xQ.val - z‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (Ne.symm hzx)) |
| have ht : 0 < t := div_pos hε (mul_pos (by norm_num) hnorm) |
| let y : affineSpan ℝ S := ⟨xQ.val + t • (xQ.val - z), by |
| simpa only [vsub_eq_sub, vadd_eq_add, add_comm] using |
| (affineSpan ℝ S).smul_vsub_vadd_mem t xQ.property |
| (subset_affineSpan ℝ S hz) xQ.property⟩ |
| have hyball : y ∈ Metric.ball xQ ε := by |
| change dist (xQ.val + t • (xQ.val - z)) xQ.val < ε |
| rw [dist_eq_norm, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_pos ht] |
| have heq : t * ‖xQ.val - z‖ = ε / 2 := by |
| dsimp [t] |
| field_simp |
| rw [heq] |
| linarith |
| have hy : y.val ∈ S := hball hyball |
| have hle := hmax y.val hy |
| change ⟪v, xQ.val + t • (xQ.val - z)⟫ ≤ ⟪v, xQ.val⟫ at hle |
| rw [inner_add_right, real_inner_smul_right, inner_sub_right] at hle |
| exact le_antisymm (hmax z hz) (by nlinarith) |
|
|
| namespace HPolytope |
|
|
| variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) |
|
|
| theorem exists_active_of_mem_exposedFace {u x : Euclidean d} (hu : u ≠ 0) |
| (hx : x ∈ exposedFace P.body u) : ∃ i, x ∈ P.face i := by |
| obtain ⟨t, ht, hxt, i, _, hi⟩ := P.exists_front_point x u hu hx.1 |
| have hmax := hx.2 (x + t • u) hxt |
| change ⟪u, x + t • u⟫ ≤ ⟪u, x⟫ at hmax |
| rw [inner_add_right, real_inner_smul_right] at hmax |
| have hpos := real_inner_self_pos.mpr hu |
| have ht0 : t = 0 := by nlinarith |
| refine ⟨i, hx.1, ?_⟩ |
| change P.offset i - ⟪P.normal i, x + t • u⟫ = 0 at hi |
| simp only [ht0, zero_smul, add_zero] at hi |
| exact (sub_eq_zero.mp hi).symm |
|
|
| theorem exposedFace_eq_face {u : Euclidean d} (hu : u ≠ 0) |
| (hdim : Module.finrank ℝ (vectorSpan ℝ (exposedFace P.body u)) = d - 1) : |
| ∃ i, exposedFace P.body u = P.face i := by |
| have hconv : Convex ℝ (exposedFace P.body u) := |
| (show IsExposed ℝ P.body (exposedFace P.body u) from |
| ContinuousLinearMap.toExposed.isExposed).convex P.body_convex |
| have hne := exposedFace_nonempty P.body P.compact P.body_nonempty u |
| obtain ⟨x, hx⟩ := (intrinsicInterior_nonempty hconv).mpr hne |
| have hxF : x ∈ exposedFace P.body u := intrinsicInterior_subset hx |
| obtain ⟨i, hxi⟩ := P.exists_active_of_mem_exposedFace hu hxF |
| have hinner : ∀ z ∈ exposedFace P.body u, ⟪P.normal i, z⟫ = P.offset i := by |
| intro z hz |
| apply Eq.trans (inner_eq_of_mem_intrinsicInterior hx (P.normal i) ?_ z hz) hxi.2 |
| intro y hy |
| rw [hxi.2] |
| exact hy.1 i |
| have hle : vectorSpan ℝ (exposedFace P.body u) ≤ normalHyperplane (P.normal i) := by |
| rw [vectorSpan_def] |
| apply Submodule.span_le.mpr |
| rintro _ ⟨a, ha, b, hb, rfl⟩ |
| apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr |
| change ⟪P.normal i, a - b⟫ = 0 |
| rw [inner_sub_right, hinner a ha, hinner b hb, sub_self] |
| have hn : P.normal i ≠ 0 := by |
| intro h |
| have hnorm := P.normal_unit i |
| rw [h, norm_zero] at hnorm |
| norm_num at hnorm |
| have hspan_i : vectorSpan ℝ (exposedFace P.body u) = normalHyperplane (P.normal i) := |
| Submodule.eq_of_le_of_finrank_eq hle (hdim.trans (normalHyperplane_finrank _ hn).symm) |
| have hspan_u : vectorSpan ℝ (exposedFace P.body u) = normalHyperplane u := |
| Submodule.eq_of_le_of_finrank_eq (exposedFace_vectorSpan_le_normalHyperplane P.body u) |
| (hdim.trans (normalHyperplane_finrank u hu).symm) |
| refine ⟨i, Subset.antisymm (fun z hz => ⟨hz.1, hinner z hz⟩) ?_⟩ |
| intro y hy |
| have hyi : y - x ∈ normalHyperplane (P.normal i) := by |
| apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr |
| rw [inner_sub_right, hy.2, hxi.2, sub_self] |
| have hyu : y - x ∈ normalHyperplane u := by |
| rwa [← hspan_u, hspan_i] |
| have heq : ⟪u, y⟫ = ⟪u, x⟫ := by |
| have h := Submodule.mem_orthogonal_singleton_iff_inner_right.mp hyu |
| simpa only [inner_sub_right, sub_eq_zero] using h |
| refine ⟨hy.1, ?_⟩ |
| intro z hz |
| change ⟪u, z⟫ ≤ ⟪u, y⟫ |
| rw [heq] |
| exact hxF.2 z hz |
|
|
| end HPolytope |
| end Paper092 |
|
|
| end |
|
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| end OAI |
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|
