| 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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|
