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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 | import OAI.Geometry.ProjectionVolume.ExposedProduct
namespace OAI
noncomputable section
open Set
open scoped RealInnerProductSpace
namespace Paper092
theorem exposedFace_vectorSpan_le_normalHyperplane {n : ℕ}
(K : Set (Euclidean n)) (u : Euclidean n) :
vectorSpan ℝ (exposedFace K u) ≤ normalHyperplane u := by
rw [vectorSpan_def]
apply Submodule.span_le.mpr
rintro _ ⟨x, hx, y, hy, rfl⟩
apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr
change ⟪u, x - y⟫ = 0
rw [inner_sub_right]
exact sub_eq_zero.mpr (le_antisymm (hy.2 x hx.1) (hx.2 y hy.1))
theorem exposedFace_finrank_le {n : ℕ} (K : Set (Euclidean n))
{u : Euclidean n} (hu : u ≠ 0) :
Module.finrank ℝ (vectorSpan ℝ (exposedFace K u)) ≤ n - 1 := by
have h := Submodule.finrank_mono (exposedFace_vectorSpan_le_normalHyperplane K u)
rwa [normalHyperplane_finrank u hu] at h
theorem exposedFace_ssubset {n : ℕ} (K : Set (Euclidean n))
(hint : (interior K).Nonempty) {u : Euclidean n} (hu : u ≠ 0) :
exposedFace K u ⊂ K := by
refine ssubset_iff_subset_ne.mpr ⟨fun _ hx => hx.1, ?_⟩
intro heq
obtain ⟨x, hx⟩ := hint
obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp hx)
have hnorm : 0 < ‖u‖ := norm_pos_iff.mpr hu
let t : ℝ := ε / (2 * ‖u‖)
have ht : 0 < t := div_pos hε (mul_pos (by norm_num) hnorm)
have hdist : dist (x + t • u) x < ε := by
rw [dist_eq_norm, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_pos ht]
dsimp [t]
field_simp
linarith
have hy := hball (Metric.mem_ball.mpr hdist)
have hface : x ∈ exposedFace K u := heq.symm ▸ interior_subset hx
have hle := hface.2 (x + t • u) hy
change ⟪u, x + t • u⟫ ≤ ⟪u, x⟫ at hle
rw [inner_add_right, real_inner_smul_right, real_inner_self_eq_norm_sq] at hle
have hpos := mul_pos ht (sq_pos_of_pos hnorm)
linarith
end Paper092
end
end OAI
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