File size: 2,060 Bytes
cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import OAI.Geometry.ProjectionVolume.FacetExhaustiveness
import OAI.Geometry.ProjectionVolume.FacetUniqueness
import OAI.Geometry.ProjectionVolume.ProductFacetFamilies
namespace OAI
universe uι
noncomputable section
open Set
open scoped RealInnerProductSpace
namespace Paper092.HPolytope
variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι)
theorem face_eq_exposedFace (i : ι) (hne : (P.face i).Nonempty) :
P.face i = exposedFace P.body (P.normal i) := by
ext x
constructor
· rintro ⟨hx, hxi⟩
refine ⟨hx, ?_⟩
intro y hy
change ⟪P.normal i, y⟫ ≤ ⟪P.normal i, x⟫
rw [hxi]
exact hy i
· intro hx
obtain ⟨y, hy⟩ := hne
refine ⟨hx.1, le_antisymm (hx.1 i) ?_⟩
have h := hx.2 y hy.1
change ⟪P.normal i, y⟫ ≤ ⟪P.normal i, x⟫ at h
rwa [hy.2] at h
theorem face_isExposed (i : ι) : IsExposed ℝ P.body (P.face i) := by
intro hne
refine ⟨innerSL ℝ (P.normal i), ?_⟩
exact P.face_eq_exposedFace i hne
theorem exposed_facet_eq_unique_face (hd : 0 < d) (F : Set (Euclidean d))
(hF : IsExposed ℝ P.body F) (hne : F.Nonempty)
(hdim : Module.finrank ℝ (vectorSpan ℝ F) = d - 1) :
∃! i, F = P.face i := by
obtain ⟨u, rfl⟩ := eq_exposedFace_of_isExposed hF hne
have hu : u ≠ 0 := by
intro hzero
rw [hzero, exposedFace_zero,
vectorSpan_finrank_of_interior_nonempty P.body P.body_interior_nonempty] at hdim
omega
obtain ⟨i, hi⟩ := P.exposedFace_eq_face hu hdim
refine ⟨i, hi, ?_⟩
intro j hj
apply P.face_index_unique_of_finrank j i
· rwa [← hj]
· rwa [← hj]
· exact hj.symm.trans hi
theorem finite_exposed_facets (hd : 0 < d) :
{F : Set (Euclidean d) | IsExposed ℝ P.body F ∧ F.Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ F) = d - 1}.Finite := by
apply (Set.finite_range P.face).subset
rintro F ⟨hF, hne, hdim⟩
obtain ⟨i, hi, _⟩ := P.exposed_facet_eq_unique_face hd F hF hne hdim
exact ⟨i, hi.symm⟩
end Paper092.HPolytope
end
end OAI
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