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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 | import OAI.Geometry.ProjectionVolume.Basic
import OAI.Geometry.ProjectionVolume.SupportGeometry
import OAI.Geometry.ProjectionVolume.Brightness
import Mathlib.Analysis.Normed.Affine.Convex
namespace OAI
open Set Metric Filter
open scoped Pointwise Topology RealInnerProductSpace
namespace Paper092
theorem convex_smul_subset_interior {n : ℕ} {K : Set (Euclidean n)}
(hK : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K)
{r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) : r • K ⊆ interior K := by
rintro _ ⟨x, hx, rfl⟩
simpa only [smul_zero, add_zero] using
hK.combo_self_interior_mem_interior hx h0 hr (sub_pos.mpr hr1) (add_sub_cancel _ _)
theorem exists_inscribed_finset_sandwich {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (hv : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K)
{r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) :
∃ t : Finset (Euclidean n),
r • K ⊆ interior (convexHull ℝ (t : Set (Euclidean n))) ∧
convexHull ℝ (t : Set (Euclidean n)) ⊆ K ∧
(0 : Euclidean n) ∈ interior (convexHull ℝ (t : Set (Euclidean n))) := by
obtain ⟨t, ht, htK⟩ := (hv.smul r).exists_subset_interior_convexHull_finset_of_isCompact
(hc.smul r) (mem_nhdsSet_iff_exists.mpr
⟨interior K, isOpen_interior, convex_smul_subset_interior hv h0 hr hr1, interior_subset⟩)
refine ⟨t, ht, htK, ht ?_⟩
exact ⟨0, interior_subset h0, smul_zero r⟩
theorem support_mono {n : ℕ} {A B : Set (Euclidean n)}
(hne : A.Nonempty) (hc : IsCompact B) (h : A ⊆ B) (u : Euclidean n) :
SupportGeometry.support A (innerSL ℝ u) ≤
SupportGeometry.support B (innerSL ℝ u) :=
SupportGeometry.support_le hne _ fun _ hx => SupportGeometry.le_support hc _ (h hx)
theorem projectionBody_support_le {n : ℕ} (K : Set (Euclidean n)) (u : Euclidean n) :
SupportGeometry.support (projectionBody K) (innerSL ℝ u) ≤ brightness K u :=
SupportGeometry.support_le ⟨0, zero_mem_projectionBody K⟩ _ fun _ hx => hx u
end Paper092
end OAI
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