| import OAI.Geometry.ProjectionVolume.PolytopeFacets | |
| import OAI.Geometry.ProjectionVolume.CubeFibers | |
| import Mathlib.Analysis.Convex.Intrinsic | |
| import Mathlib.Analysis.Convex.Measure | |
| namespace OAI | |
| universe uι | |
| open Set MeasureTheory | |
| open scoped RealInnerProductSpace | |
| noncomputable section | |
| namespace Paper092.HPolytope | |
| variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) | |
| theorem projectionVolume_le_surface (S : Set (Euclidean d)) (u : Euclidean d) (hu : u ≠ 0) : | |
| projectionVolume S u ≤ μHE[d - 1] S := by | |
| unfold projectionVolume | |
| rw [← InnerProductSpace.euclideanHausdorffMeasure_eq_volume, | |
| normalHyperplane_finrank u hu] | |
| simp only [Measure.euclideanHausdorffMeasure_def, Measure.smul_apply] | |
| simp only [ENNReal.smul_def, smul_eq_mul] | |
| exact mul_le_mul_right | |
| (hausdorffMeasure_orthogonalProjectionOnto_le (normalHyperplane u) | |
| ((d - 1 : ℕ) : ℝ) S (by positivity)) _ | |
| theorem projected_face_inter_null (u : Euclidean d) (hu : u ≠ 0) (a b : ι) (hab : a ≠ b) : | |
| volume ((normalHyperplane u).orthogonalProjectionOnto '' (P.face a ∩ P.face b)) = 0 := by | |
| apply le_antisymm _ bot_le | |
| exact (projectionVolume_le_surface (P.face a ∩ P.face b) u hu).trans_eq | |
| (P.face_inter_surface_null a b hab) | |
| theorem strict_face_point_mem_intrinsicInterior (a : ι) (x : Euclidean d) | |
| (hx : x ∈ P.face a) (hstrict : ∀ b, b ≠ a → ⟪P.normal b, x⟫ < P.offset b) : | |
| x ∈ intrinsicInterior ℝ (P.face a) := by | |
| classical | |
| let Q := affineSpan ℝ (P.face a) | |
| let T : AffineSubspace ℝ (Euclidean d) := | |
| (affineSpan ℝ ({P.offset a} : Set ℝ)).comap (innerₛₗ ℝ (P.normal a)).toAffineMap | |
| have hQle : Q ≤ T := by | |
| apply affineSpan_le.mpr | |
| intro y hy | |
| simpa [T] using hy.2 | |
| have hplane (y : Q) : ⟪P.normal a, (y : Euclidean d)⟫ = P.offset a := by | |
| have h := hQle y.property | |
| simpa [T] using h | |
| let S : Set Q := ⋂ b : {b : ι // b ≠ a}, | |
| {y | ⟪P.normal b.val, (y : Euclidean d)⟫ < P.offset b.val} | |
| have hopen : IsOpen S := isOpen_iInter_of_finite fun b => | |
| isOpen_lt ((innerSL ℝ (P.normal b.val)).continuous.comp continuous_subtype_val) continuous_const | |
| let xQ : Q := ⟨x, subset_affineSpan ℝ _ hx⟩ | |
| have hxS : xQ ∈ S := mem_iInter.mpr (fun b => hstrict b.val b.property) | |
| have hsub : S ⊆ (Subtype.val ⁻¹' P.face a : Set Q) := by | |
| intro y hy | |
| refine ⟨?_, hplane y⟩ | |
| intro b | |
| by_cases hba : b = a | |
| · simpa only [hba, hplane y] using le_refl (P.offset a) | |
| · exact (mem_iInter.mp hy ⟨b, hba⟩).le | |
| apply mem_intrinsicInterior.mpr | |
| refine ⟨xQ, interior_mono hsub ?_, rfl⟩ | |
| simpa only [hopen.interior_eq] using hxS | |
| def badProjection (u : Euclidean d) : Set (normalHyperplane u) := | |
| ⋃ a : ι, ⋃ b : {b : ι // b ≠ a}, | |
| (normalHyperplane u).orthogonalProjectionOnto '' (P.face a ∩ P.face b.val) | |
| theorem badProjection_null (u : Euclidean d) (hu : u ≠ 0) : volume (P.badProjection u) = 0 := by | |
| apply measure_iUnion_null | |
| intro a | |
| apply measure_iUnion_null | |
| intro b | |
| exact P.projected_face_inter_null u hu a b.val (Ne.symm b.property) | |
| theorem strict_of_not_badProjection (u : Euclidean d) (y : normalHyperplane u) | |
| (hy : y ∉ P.badProjection u) (a : ι) (x : Euclidean d) (hx : x ∈ P.face a) | |
| (hproj : (normalHyperplane u).orthogonalProjectionOnto x = y) : | |
| ∀ b, b ≠ a → ⟪P.normal b, x⟫ < P.offset b := by | |
| intro b hba | |
| apply lt_of_le_of_ne (hx.1 b) | |
| intro hbx | |
| apply hy | |
| exact mem_iUnion.mpr ⟨a, mem_iUnion.mpr ⟨⟨b, hba⟩, x, ⟨hx, hx.1, hbx⟩, hproj⟩⟩ | |
| theorem affineSpan_face_of_not_badProjection (u : Euclidean d) | |
| (y : normalHyperplane u) (hy : y ∉ P.badProjection u) (a : ι) | |
| (ha : y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face a) : | |
| (affineSpan ℝ (P.face a) : Set (Euclidean d)) = | |
| {x | ⟪P.normal a, x⟫ = P.offset a} := by | |
| obtain ⟨x, hx, hxy⟩ := ha | |
| apply P.affineSpan_face_of_essential a | |
| intro hred | |
| obtain ⟨b, hba, hb⟩ := P.exists_other_active_of_redundant a hred x hx | |
| exact (ne_of_lt (P.strict_of_not_badProjection u y hy a x hx hxy b hba)) hb | |
| theorem parameter_of_projection_eq (u : Euclidean d) (y : normalHyperplane u) | |
| (x : Euclidean d) (hx : (normalHyperplane u).orthogonalProjectionOnto x = y) : | |
| ∃ t : ℝ, x = (y : Euclidean d) + t • u := by | |
| apply (Paper092.projection_eq_iff_parallel u x y).mp | |
| exact hx.trans ((normalHyperplane u).orthogonalProjectionOnto_mem_subspace_eq_self y).symm | |
| theorem interval_of_opposite_faces (u y : Euclidean d) (a b : ι) (l r : ℝ) | |
| (hla : y + l • u ∈ P.face a) (hrb : y + r • u ∈ P.face b) | |
| (ha : 0 < P.velocity u a) (hb : P.velocity u b < 0) : | |
| lineFiber P.body y u = Icc l r := by | |
| have hl := (P.mem_face_iff_slack a _).mp hla | |
| have hr := (P.mem_face_iff_slack b _).mp hrb | |
| rw [P.slack_add_smul] at hl hr | |
| ext t | |
| change y + t • u ∈ P.body ↔ l ≤ t ∧ t ≤ r | |
| constructor | |
| · intro ht | |
| have hta := (P.mem_body_iff_slack _).mp ht a | |
| have htb := (P.mem_body_iff_slack _).mp ht b | |
| rw [P.slack_add_smul] at hta htb | |
| constructor <;> nlinarith | |
| · rintro ⟨hlt, htr⟩ | |
| apply (P.mem_body_iff_slack _).mpr | |
| intro i | |
| rw [P.slack_add_smul] | |
| by_cases hi : 0 ≤ P.velocity u i | |
| · have hli := (P.mem_body_iff_slack _).mp hl.1 i | |
| rw [P.slack_add_smul] at hli | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hlt) hi] | |
| · have hri := (P.mem_body_iff_slack _).mp hr.1 i | |
| rw [P.slack_add_smul] at hri | |
| nlinarith [mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr htr) (le_of_not_ge hi)] | |
| theorem two_facet_interiors_outside_bad (u : Euclidean d) (hu : u ≠ 0) | |
| (y : normalHyperplane u) (hy : y ∉ P.badProjection u) | |
| (hyP : y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.body) : | |
| ∃ (a b : ι) (l r : ℝ), a ≠ b ∧ l < r ∧ | |
| lineFiber P.body (y : Euclidean d) u = Icc l r ∧ | |
| (y : Euclidean d) + l • u ∈ intrinsicInterior ℝ (P.face a) ∧ | |
| (y : Euclidean d) + r • u ∈ intrinsicInterior ℝ (P.face b) ∧ | |
| {i | y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face i} = {a, b} ∧ | |
| {i | y ∈ (normalHyperplane u).orthogonalProjectionOnto '' | |
| intrinsicInterior ℝ (P.face i)} = {a, b} := by | |
| classical | |
| obtain ⟨p, hp, hpy⟩ := hyP | |
| obtain ⟨tf, _, hpf, b, hb, hqf⟩ := P.exists_front_point p u hu hp | |
| have hf : p + tf • u ∈ P.face b := (P.mem_face_iff_slack _ _).mpr ⟨hpf, hqf⟩ | |
| have hfp : (normalHyperplane u).orthogonalProjectionOnto (p + tf • u) = y := | |
| ((Paper092.projection_eq_iff_parallel u (p + tf • u) p).mpr ⟨tf, rfl⟩).trans hpy | |
| obtain ⟨r, hrf⟩ := parameter_of_projection_eq u y (p + tf • u) hfp | |
| have hrb : (y : Euclidean d) + r • u ∈ P.face b := by rwa [hrf] at hf | |
| have hrp : (normalHyperplane u).orthogonalProjectionOnto ((y : Euclidean d) + r • u) = y := by | |
| rwa [hrf] at hfp | |
| obtain ⟨tb, _, hpb, a, ha', hqb⟩ := P.exists_front_point p (-u) (neg_ne_zero.mpr hu) hp | |
| have ha : 0 < P.velocity u a := by simpa only [velocity_neg, neg_lt_zero] using ha' | |
| have hback : p + tb • (-u) ∈ P.face a := (P.mem_face_iff_slack _ _).mpr ⟨hpb, hqb⟩ | |
| have hbp : (normalHyperplane u).orthogonalProjectionOnto (p + tb • (-u)) = y := by | |
| have heq : p + tb • (-u) = p + (-tb) • u := by simp | |
| rw [heq] | |
| exact ((Paper092.projection_eq_iff_parallel u (p + (-tb) • u) p).mpr ⟨-tb, rfl⟩).trans hpy | |
| obtain ⟨l, hlb⟩ := parameter_of_projection_eq u y (p + tb • (-u)) hbp | |
| have hla : (y : Euclidean d) + l • u ∈ P.face a := by rwa [hlb] at hback | |
| have hlp : (normalHyperplane u).orthogonalProjectionOnto ((y : Euclidean d) + l • u) = y := by | |
| rwa [hlb] at hbp | |
| have hab : a ≠ b := by intro h; subst b; linarith | |
| have hls := P.strict_of_not_badProjection u y hy a _ hla hlp | |
| have hrs := P.strict_of_not_badProjection u y hy b _ hrb hrp | |
| have hlr : l < r := by | |
| have hpos : 0 < P.slack ((y : Euclidean d) + l • u) b := sub_pos.mpr (hls b hab.symm) | |
| have hzero := ((P.mem_face_iff_slack b _).mp hrb).2 | |
| rw [P.slack_add_smul] at hpos hzero | |
| by_contra h | |
| have hprod := mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (le_of_not_gt h)) hb.le | |
| nlinarith | |
| have hint := P.interval_of_opposite_faces u y a b l r hla hrb ha hb | |
| have hli := P.strict_face_point_mem_intrinsicInterior a _ hla hls | |
| have hri := P.strict_face_point_mem_intrinsicInterior b _ hrb hrs | |
| have hfaces : {i | y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face i} = {a, b} := by | |
| ext i | |
| change (y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face i) ↔ i = a ∨ i = b | |
| constructor | |
| · rintro ⟨z, hz, hzy⟩ | |
| by_cases hia : i = a | |
| · exact Or.inl hia | |
| by_cases hib : i = b | |
| · exact Or.inr hib | |
| exfalso | |
| obtain ⟨t, hzt⟩ := parameter_of_projection_eq u y z hzy | |
| have ht : t ∈ Icc l r := by | |
| rw [← hint] | |
| change (y : Euclidean d) + t • u ∈ P.body | |
| rw [← hzt] | |
| exact hz.1 | |
| have hzero : P.slack ((y : Euclidean d) + t • u) i = 0 := by | |
| rw [← hzt] | |
| exact ((P.mem_face_iff_slack i z).mp hz).2 | |
| have hlpos : 0 < P.slack ((y : Euclidean d) + l • u) i := sub_pos.mpr (hls i hia) | |
| have hrpos : 0 < P.slack ((y : Euclidean d) + r • u) i := sub_pos.mpr (hrs i hib) | |
| rw [P.slack_add_smul] at hzero hlpos hrpos | |
| by_cases hi : 0 ≤ P.velocity u i | |
| · nlinarith [mul_nonneg (sub_nonneg.mpr ht.1) hi] | |
| · nlinarith [mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr ht.2) (le_of_not_ge hi)] | |
| · rintro (rfl | rfl) | |
| · exact ⟨_, hla, hlp⟩ | |
| · exact ⟨_, hrb, hrp⟩ | |
| refine ⟨a, b, l, r, hab, hlr, hint, hli, hri, hfaces, ?_⟩ | |
| ext i | |
| change (y ∈ (normalHyperplane u).orthogonalProjectionOnto '' intrinsicInterior ℝ (P.face i)) ↔ | |
| i = a ∨ i = b | |
| constructor | |
| · intro hi | |
| have hiface : y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face i := | |
| Set.image_mono intrinsicInterior_subset hi | |
| have hm : i ∈ ({a, b} : Set ι) := hfaces ▸ hiface | |
| simpa only [mem_insert_iff, mem_singleton_iff] using hm | |
| · rintro (rfl | rfl) | |
| · exact ⟨_, hli, hlp⟩ | |
| · exact ⟨_, hri, hrp⟩ | |
| theorem ae_two_facet_interiors (u : Euclidean d) (hu : u ≠ 0) : | |
| ∀ᵐ y : normalHyperplane u ∂volume, | |
| y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.body → | |
| ∃ (a b : ι) (l r : ℝ), a ≠ b ∧ l < r ∧ | |
| lineFiber P.body (y : Euclidean d) u = Icc l r ∧ | |
| (y : Euclidean d) + l • u ∈ intrinsicInterior ℝ (P.face a) ∧ | |
| (y : Euclidean d) + r • u ∈ intrinsicInterior ℝ (P.face b) ∧ | |
| {i | y ∈ (normalHyperplane u).orthogonalProjectionOnto '' P.face i} = {a, b} ∧ | |
| {i | y ∈ (normalHyperplane u).orthogonalProjectionOnto '' | |
| intrinsicInterior ℝ (P.face i)} = {a, b} := by | |
| have hgood : ∀ᵐ y : normalHyperplane u ∂volume, y ∉ P.badProjection u := by | |
| rw [ae_iff] | |
| simpa only [not_not, Set.ofPred_mem_eq] using P.badProjection_null u hu | |
| filter_upwards [hgood] with y hy | |
| exact P.two_facet_interiors_outside_bad u hu y hy | |
| end Paper092.HPolytope | |
| end | |
| end OAI | |