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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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Geometry.ProjectionVolume.FacetUniqueness
import OAI.Geometry.ProjectionVolume.ProjectionNullity
namespace OAI
universe uι
open Set MeasureTheory
open scoped RealInnerProductSpace Classical Pointwise
noncomputable section
namespace Paper092.HPolytope
variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι)
theorem face_finrank_le (i : ι) :
Module.finrank ℝ (vectorSpan ℝ (P.face i)) ≤ d - 1 := by
have hn : P.normal i ≠ 0 := by
intro h
have := P.normal_unit i
simp only [h, norm_zero] at this
norm_num at this
have h := Submodule.finrank_mono (P.face_vectorSpan_le_normalHyperplane i)
rwa [normalHyperplane_finrank (P.normal i) hn] at h
theorem face_surface_zero_of_finrank_lt (i : ι)
(h : Module.finrank ℝ (vectorSpan ℝ (P.face i)) < d - 1) :
μHE[d - 1] (P.face i) = 0 := by
have hn : P.normal i ≠ 0 := by
intro h
have := P.normal_unit i
simp only [h, norm_zero] at this
norm_num at this
have hz := projectionVolume_eq_zero_of_finrank_lt (P.face i) hn h
rw [projectionVolume_of_subset_affine_hyperplane (P.normal i) (P.normal i)
(P.normal_unit i) (P.normal_unit i) (P.offset i) (P.face i) (fun _ hx => hx.2)] at hz
simpa only [real_inner_self_eq_norm_sq, P.normal_unit, one_pow, abs_one,
ENNReal.ofReal_one, one_mul] using hz
theorem faceArea_zero_of_not_facet (i : ι)
(h : ¬ ((P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1)) : P.faceArea i = 0 := by
by_cases hne : (P.face i).Nonempty
· have hdim : Module.finrank ℝ (vectorSpan ℝ (P.face i)) ≠ d - 1 :=
fun heq => h ⟨hne, heq⟩
rw [faceArea, P.face_surface_zero_of_finrank_lt i (lt_of_le_of_ne (P.face_finrank_le i) hdim),
ENNReal.toReal_zero]
· have he : P.face i = ∅ := by
ext x
constructor
· intro hx
exact (hne ⟨x, hx⟩).elim
· intro hx
exact hx.elim
simp only [faceArea, he, measure_empty, ENNReal.toReal_zero]
theorem brightness_eq_facet_sum (u : Euclidean d) :
brightness P.body u = (1 / 2 : ℝ) *
∑ i ∈ Finset.univ.filter (fun i => (P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1),
P.faceArea i * |⟪P.normal i, u⟫| := by
have hsum : (∑ i, P.faceArea i * |⟪u, P.normal i⟫|) =
∑ i ∈ Finset.univ.filter (fun i => (P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1),
P.faceArea i * |⟪u, P.normal i⟫| := by
symm
apply Finset.sum_subset (Finset.filter_subset _ _)
intro i hi hni
have hn : ¬ ((P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1) :=
fun hf => hni (Finset.mem_filter.mpr ⟨hi, hf⟩)
rw [P.faceArea_zero_of_not_facet i hn, zero_mul]
rw [P.brightness_eq_sum, hsum]
simp_rw [real_inner_comm u]
ring
theorem projectionBody_eq_facet_sum :
projectionBody P.body =
∑ i ∈ Finset.univ.filter (fun i => (P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1),
segment ℝ (-(P.faceArea i / 2) • P.normal i) ((P.faceArea i / 2) • P.normal i) := by
rw [P.projectionBody_eq_facetZonotope]
unfold facetZonotope
symm
apply Finset.sum_subset (Finset.filter_subset _ _)
intro i hi hni
have hn : ¬ ((P.face i).Nonempty ∧
Module.finrank ℝ (vectorSpan ℝ (P.face i)) = d - 1) :=
fun hf => hni (Finset.mem_filter.mpr ⟨hi, hf⟩)
simp only [P.faceArea_zero_of_not_facet i hn, zero_div, neg_zero, zero_smul,
segment_same, Set.singleton_zero]
end Paper092.HPolytope
end
end OAI
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