| import Mathlib |
| import OAI.Geometry.Borsuk.Support.BalancedChart |
| import OAI.Geometry.Borsuk.LocalParity.RegularBase |
| import OAI.Geometry.Borsuk.MatrixGeometry.BlockLine |
| import OAI.Geometry.Borsuk.LocalParity.RegularPatch |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| namespace BorsukNine |
| noncomputable def raySet {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (y : E) : Set E := |
| {v | v = (‖v‖ / ‖y‖) • y} |
| lemma raySet_isClosed {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (y : E) : |
| IsClosed (raySet y) := isClosed_eq continuous_id ((continuous_norm.div_const _).smul continuous_const) |
| lemma mem_raySet_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {v y : E} (hv : v ≠ 0) (hy : y ≠ 0) : v ∈ raySet y ↔ ∃ d : ℝ, 0 < d ∧ v = d • y := by |
| constructor |
| · intro h |
| exact ⟨‖v‖ / ‖y‖,div_pos (norm_pos_iff.mpr hv) (norm_pos_iff.mpr hy),h⟩ |
| · rintro ⟨d,hd,rfl⟩ |
| change d • y = (‖d • y‖ / ‖y‖) • y |
| rw [norm_smul,Real.norm_eq_abs,abs_of_pos hd,mul_div_cancel_right₀ _ (norm_ne_zero_iff.mpr hy)] |
| end BorsukNine |
|
|
| namespace BorsukNine |
| open Set |
| open scoped MatrixOrder Matrix.Norms.L2Operator RealInnerProductSpace |
| section MatrixFibres |
| variable {k : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι] |
| local notation "E" => EuclideanSpace ℝ (Fin (k+1)) |
| local notation "W" => EuclideanSpace ℝ (Fin k) |
| local notation "M" => Matrix (Fin (k+1)) (Fin (k+1)) ℝ |
| local notation "T" => traceSphere (Fin (k+1)) |
| local notation "Y" => EuclideanSpace ℝ ι |
| local notation "op" => Matrix.toEuclideanCLM (n := Fin (k+1)) (𝕜 := ℝ) |
| noncomputable def matrixRayFibre (F : T → Y) (y : Y) (c : E) : Set T := |
| {A | F A ∈ raySet y ∧ A.val⁺ = Matrix.trace A.val⁺ • rankOne c} |
| omit [DecidableEq ι] in |
| lemma matrixRayFibre_closed (F : T → Y) (hF : Continuous F) (y : Y) (c : E) : |
| IsClosed (matrixRayFibre F y c) := by |
| have hp : Continuous (fun A : T => A.val⁺) := |
| matrix_posPart_continuousOn.comp_continuous continuous_subtype_val (fun A => A.property.1) |
| exact ((raySet_isClosed y).preimage hF).inter |
| (isClosed_eq hp ((matrix_trace_continuous.comp hp).smul continuous_const)) |
| lemma Admissible.facet_support {f : E → ι → ℝ} (_hf : Admissible f) |
| {I : Finset ι} (a : Fin (k+1) ≃ ↥I) {z : W} (hz : z ∈ simplexInterior k) |
| {c : E} (hc : f c = facetPoint a z) : positiveSupport f c = I := by |
| ext i |
| rw [mem_positiveSupport,hc] |
| by_cases hi : i ∈ I |
| · exact iff_of_true (facetPoint_pos a hz i hi) hi |
| · rw [facetPoint_zero a z hi] |
| simp [hi] |
| omit [DecidableEq ι] in |
| lemma matrixRayFibre_representative {F : T → Y} {y : Y} {w c : E} (hc : ‖c‖ = 1) |
| (hwc : 0 < ⟪w,c⟫) {A : T} (hline : HasPositiveLine A.val) |
| (hK : A ∈ matrixRayFibre F y c) : lineRepresentative w A.val = c := |
| lineRepresentative_split hc (hasPositiveLine_trace_pos hline) hK.2 hwc |
| omit [DecidableEq ι] in |
| lemma Admissible.matrixRayFibre_subset_patch {f : E → ι → ℝ} (_hf : Admissible f) |
| {F : T → Y} {y : Y} {w c : E} (hc : ‖c‖ = 1) (hwc : 0 < ⟪w,c⟫) |
| {I : Finset ι} (hp : ∀ i ∈ I, 0 < f c i) |
| (e : W →ₗᵢ[ℝ] E) (he : ∀ v, ⟪c,e v⟫ = 0) |
| (hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u) {δ : ℝ} (hδ : 0 < δ) |
| {A : T} (hK : A ∈ matrixRayFibre F y c) |
| (hU : IsUnit A.val) (hline : HasPositiveLine A.val) (hAw : op A.val⁺ w ≠ 0) : |
| A ∈ regularPatch f I w c e δ := by |
| have hr := matrixRayFibre_representative hc hwc hline hK |
| have ht := hasPositiveLine_trace_pos hline |
| refine ⟨hU,hline,hAw,?_,?_,?_,?_,?_⟩ |
| · simpa only [hr] using hp |
| · rw [hr,real_inner_self_eq_norm_sq,hc]; norm_num |
| · rw [hr,hemisphereInverse,adjoint_normal c e he,smul_zero,norm_zero] |
| exact hδ |
| · rw [blockScalar_positiveLine A.property.1 hc ht hK.2] |
| exact ht |
| · exact blockCompress_negative e A.property.1 hU hc he hes ht hK.2 |
| end MatrixFibres |
| end BorsukNine |
|
|
| end OAI |
|
|
