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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 | import OAI.Geometry.ProjectionVolume.PolytopeDefinitions
import Mathlib.Analysis.Convex.KreinMilman
import Mathlib.Tactic.FunProp
import Mathlib.Tactic.Linarith
namespace OAI
universe uι
open Set Filter Topology
open scoped RealInnerProductSpace
namespace Paper092
namespace HPolytope
variable {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι)
theorem extreme_active_unique {x y : Euclidean d}
(hx : x ∈ P.body.extremePoints ℝ)
(ha : ∀ i, ⟪P.normal i, x⟫ = P.offset i → ⟪P.normal i, y⟫ = P.offset i) :
y = x := by
have hnear : ∀ᶠ t : ℝ in 𝓝 0, ∀ i,
⟪P.normal i, x⟫ < P.offset i →
⟪P.normal i, x + t • (y - x)⟫ < P.offset i := by
apply Filter.eventually_all.mpr
intro i
by_cases hi : ⟪P.normal i, x⟫ < P.offset i
· have h : ∀ᶠ t : ℝ in 𝓝 0,
⟪P.normal i, x + t • (y - x)⟫ < P.offset i :=
(isOpen_lt (by fun_prop) continuous_const).mem_nhds (by simpa using hi)
filter_upwards [h] with t ht
exact fun _ => ht
· exact Filter.Eventually.of_forall fun _ h => (hi h).elim
obtain ⟨ε, hε, hsmall⟩ := Metric.eventually_nhds_iff.mp hnear
have hpert (t : ℝ) (ht : dist t 0 < ε) : x + t • (y - x) ∈ P.body := by
intro i
rcases (hx.1 i).lt_or_eq with hi | hi
· exact (hsmall ht i hi).le
· simp [inner_add_right, inner_smul_right, inner_sub_right, hi, ha i hi]
have hp : dist (ε / 2) 0 < ε := by
rw [Real.dist_eq, sub_zero, abs_of_pos (half_pos hε)]
linarith
have hm : dist (-(ε / 2)) 0 < ε := by simpa using hp
have hxplus := hx.2 (hpert (ε / 2) hp)
(show x - (ε / 2) • (y - x) ∈ P.body by simpa only [neg_smul, ← sub_eq_add_neg] using hpert (-(ε / 2)) hm)
(mem_openSegment_add_sub (𝕜 := ℝ) x ((ε / 2) • (y - x)))
have hz : (ε / 2) • (y - x) = 0 := add_left_cancel (hxplus.trans (add_zero x).symm)
exact sub_eq_zero.mp ((smul_eq_zero.mp hz).resolve_left (ne_of_gt (half_pos hε)))
theorem finite_extremePoints : (P.body.extremePoints ℝ).Finite := by
classical
let active (x : Euclidean d) : Finset ι :=
Finset.univ.filter (fun i => ⟪P.normal i, x⟫ = P.offset i)
have hi : Set.InjOn active (P.body.extremePoints ℝ) := by
intro x hx y _ hxy
apply (P.extreme_active_unique hx ?_).symm
intro i hix
have hm : i ∈ active x := by simp [active, hix]
rw [hxy] at hm
exact (Finset.mem_filter.mp hm).2
exact (Set.toFinite (Set.univ : Set (Finset ι))).of_injOn
(fun _ _ => Set.mem_univ _) hi
theorem eq_convexHull_extremePoints : P.body = convexHull ℝ (P.body.extremePoints ℝ) := by
have hc : Convex ℝ P.body := by
change Convex ℝ {x : Euclidean d | ∀ i, ⟪P.normal i, x⟫ ≤ P.offset i}
simp only [Set.ofPred_forall]
exact convex_iInter fun i => (convex_Iic (P.offset i)).linear_preimage
(innerSL ℝ (P.normal i)).toLinearMap
have h := closure_convexHull_extremePoints P.compact hc
change closure (convexHull ℝ (P.body.extremePoints ℝ)) = P.body at h
rw [P.finite_extremePoints.isClosed_convexHull ℝ |>.closure_eq] at h
exact h.symm
end HPolytope
theorem halfspaces_are_finite_convex_hull {d : ℕ} {ι : Type uι} [Fintype ι]
(P : HPolytope d ι) :
∃ S : Finset (Euclidean d), P.body = convexHull ℝ (S : Set (Euclidean d)) := by
classical
exact ⟨P.finite_extremePoints.toFinset, by simpa using P.eq_convexHull_extremePoints⟩
end Paper092
end OAI
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