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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 | import OAI.Geometry.ProjectionVolume.PolytopeFaceGeometry
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 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
end Paper092.HPolytope
end
end OAI
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