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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import OAI.Geometry.ProjectionVolume.FiniteApproximation
import OAI.Geometry.ProjectionVolume.PolytopeDefinitions
namespace OAI
open Set Metric
open scoped Pointwise RealInnerProductSpace
namespace Paper092
theorem exists_unit_separating_support {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (hne : K.Nonempty) (hv : Convex ℝ K)
{x : Euclidean n} (hx : x ∉ K) :
∃ u : Euclidean n, ‖u‖ = 1 ∧
SupportGeometry.support K (innerSL ℝ u) < ⟪u, x⟫ := by
have hx' := (SupportGeometry.mem_iff_inner_le_support hc hne hv).not.mp hx
push Not at hx'
obtain ⟨u, hu⟩ := hx'
have hu0 : u ≠ 0 := by
intro h
have hh := (SupportGeometry.le_support hc (innerSL ℝ u) hne.choose_spec).trans_lt hu
simp [h] at hh
let v := ‖u‖⁻¹ • u
have hv1 : ‖v‖ = 1 := by simp [v, norm_smul, hu0]
obtain ⟨y, hy, hys⟩ := SupportGeometry.exists_support hc hne (innerSL ℝ v)
refine ⟨v, hv1, ?_⟩
rw [← hys]
change ⟪v, y⟫ < ⟪v, x⟫
simp only [v, real_inner_smul_left]
exact mul_lt_mul_of_pos_left ((SupportGeometry.le_support hc (innerSL ℝ u) hy).trans_lt hu)
(inv_pos.mpr (norm_pos_iff.mpr hu0))
theorem unit_support_pos_of_zero_mem_interior {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (h0 : (0 : Euclidean n) ∈ interior K)
(u : Euclidean n) (hu : ‖u‖ = 1) :
0 < SupportGeometry.support K (innerSL ℝ u) := by
obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp h0)
have hz : (r / 2) • u ∈ K := by
apply hball
simp only [mem_ball, dist_zero_right, norm_smul, Real.norm_eq_abs, hu, mul_one]
rw [abs_of_pos (half_pos hr)]
linarith
have hh := SupportGeometry.le_support hc (innerSL ℝ u) hz
simp only [innerSL_apply_apply, inner_smul_right, real_inner_self_eq_norm_sq, hu,
one_pow, mul_one] at hh
exact (half_pos hr).trans_le hh
theorem unit_support_le_of_norm_le {n : ℕ} {K : Set (Euclidean n)}
(hne : K.Nonempty) {R : ℝ} (hR : ∀ x ∈ K, ‖x‖ ≤ R)
(u : Euclidean n) (hu : ‖u‖ = 1) :
SupportGeometry.support K (innerSL ℝ u) ≤ R := by
apply SupportGeometry.support_le hne
intro x hx
exact (real_inner_le_norm u x).trans (by simpa [hu] using hR x hx)
theorem norm_le_of_basis_inner_le {n : ℕ} {x : Euclidean n} {R : ℝ}
(h : ∀ i : Fin n, |⟪EuclideanSpace.basisFun (Fin n) ℝ i, x⟫| ≤ R) :
‖x‖ ≤ n * R := by
let b := EuclideanSpace.basisFun (Fin n) ℝ
calc
‖x‖ = ‖∑ i : Fin n, ⟪b i, x⟫ • b i‖ := by rw [b.sum_repr']
_ ≤ ∑ i : Fin n, ‖⟪b i, x⟫ • b i‖ := norm_sum_le _ _
_ = ∑ i : Fin n, |⟪b i, x⟫| := by simp [norm_smul, Real.norm_eq_abs]
_ ≤ ∑ i : Fin n, R := Finset.sum_le_sum (fun i _ => h i)
_ = n * R := by simp
theorem subset_interior_smul_of_one_lt {n : ℕ} {K : Set (Euclidean n)}
(hv : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K)
{r : ℝ} (hr : 1 < r) : K ⊆ interior (r • K) := by
have heq : (AffineMap.homothety (0 : Euclidean n) r : Euclidean n → Euclidean n) =
(fun x => r • x) := by
funext x
simp [AffineMap.homothety_apply, vsub_eq_sub, vadd_eq_add]
change K ⊆ interior ((fun x : Euclidean n => r • x) '' K)
rw [← heq]
exact hv.subset_interior_image_homothety_of_one_lt h0 r hr
theorem exists_outer_hpolytope_sandwich {n : ℕ} {K : Set (Euclidean n)}
(hc : IsCompact K) (hv : Convex ℝ K) (h0 : (0 : Euclidean n) ∈ interior K)
{r : ℝ} (hr : 1 < r) :
∃ (F : Finset (Euclidean n)) (P : HPolytope n {u // u ∈ F}),
K ⊆ P.body ∧ P.body ⊆ r • K := by
classical
have hne : K.Nonempty := ⟨0, interior_subset h0⟩
obtain ⟨R, hRp, hR⟩ := hc.isBounded.exists_pos_norm_le
let b := EuclideanSpace.basisFun (Fin n) ℝ
let U := {u : Euclidean n // ‖u‖ = 1}
let V : U → Set (Euclidean n) := fun u =>
{x | SupportGeometry.support K (innerSL ℝ u.val) < ⟪u.val, x⟫}
have hKin := subset_interior_smul_of_one_lt hv h0 hr
have hcover : closedBall (0 : Euclidean n) (n * R) \ interior (r • K) ⊆ ⋃ u : U, V u := by
intro x hx
obtain ⟨u, hu, hsep⟩ := exists_unit_separating_support hc hne hv
(fun hxK => hx.2 (hKin hxK))
exact mem_iUnion.mpr ⟨⟨u, hu⟩, hsep⟩
obtain ⟨t, ht⟩ := ((isCompact_closedBall (0 : Euclidean n) (n * R)).diff
isOpen_interior).elim_finite_subcover V
(fun u => isOpen_lt continuous_const (innerSL ℝ u.val).continuous) hcover
let F := t.image Subtype.val ∪
(Finset.univ.image b ∪ Finset.univ.image (fun i : Fin n => -b i))
have htF (u : U) (hu : u ∈ t) : u.val ∈ F :=
Finset.mem_union_left _ (Finset.mem_image.mpr ⟨u, hu, rfl⟩)
have hpF (i : Fin n) : b i ∈ F := Finset.mem_union_right _
(Finset.mem_union_left _ (Finset.mem_image.mpr ⟨i, Finset.mem_univ i, rfl⟩))
have hnF (i : Fin n) : -b i ∈ F := Finset.mem_union_right _
(Finset.mem_union_right _ (Finset.mem_image.mpr ⟨i, Finset.mem_univ i, rfl⟩))
have hFunit : ∀ u ∈ F, ‖u‖ = 1 := by
intro u hu
rcases Finset.mem_union.mp hu with hu | hu
· obtain ⟨v, _, rfl⟩ := Finset.mem_image.mp hu
exact v.property
· rcases Finset.mem_union.mp hu with hu | hu
· obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hu
exact b.norm_eq_one i
· obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hu
simp
let H : Set (Euclidean n) := {x | ∀ u : {u // u ∈ F},
⟪u.val, x⟫ ≤ SupportGeometry.support K (innerSL ℝ u.val)}
have hKH : K ⊆ H := fun x hx u => SupportGeometry.le_support hc (innerSL ℝ u.val) hx
have hHbound : ∀ x ∈ H, ‖x‖ ≤ n * R := by
intro x hx
apply norm_le_of_basis_inner_le
intro i
have hp := (hx ⟨b i, hpF i⟩).trans
(unit_support_le_of_norm_le hne hR (b i) (b.norm_eq_one i))
have hn := (hx ⟨-b i, hnF i⟩).trans
(unit_support_le_of_norm_le hne hR (-b i) (by simp))
apply abs_le.mpr
refine ⟨?_, hp⟩
simpa only [inner_neg_left, neg_le] using hn
have hHr : H ⊆ r • K := by
intro x hx
by_contra hxout
have hxC : x ∈ closedBall (0 : Euclidean n) (n * R) \ interior (r • K) :=
⟨by simpa only [mem_closedBall, dist_zero_right] using hHbound x hx,
fun h => hxout (interior_subset h)⟩
obtain ⟨u, hut, hux⟩ := mem_iUnion₂.mp (ht hxC)
exact (not_lt_of_ge (hx ⟨u.val, htF u hut⟩)) hux
have hHclosed : IsClosed H := by
dsimp [H]
rw [ofPred_forall]
exact isClosed_iInter fun u => isClosed_le (innerSL ℝ u.val).continuous continuous_const
let P : HPolytope n {u // u ∈ F} := {
normal := Subtype.val
offset := fun u => SupportGeometry.support K (innerSL ℝ u.val)
normal_unit := fun u => hFunit u.val u.property
compact := (hc.smul r).of_isClosed_subset hHclosed hHr
strict_feasible := ⟨0, fun u => by
simpa only [inner_zero_right] using
unit_support_pos_of_zero_mem_interior hc h0 u.val (hFunit u.val u.property)⟩
distinct := fun i j hij => Subtype.ext (congrArg Prod.fst hij) }
exact ⟨F, P, hKH, hHr⟩
end Paper092
end OAI
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