| import OAI.Analysis.DirectCrouzeix.ExteriorMapping | |
| namespace OAI | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| noncomputable section | |
| open MeasureTheory Set Filter Metric | |
| open scoped Topology Interval ENNReal NNReal ComplexConjugate | |
| noncomputable section | |
| open Filter Metric Set | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ComplexConjugate | |
| namespace DirectCrouzeix.Geometry | |
| theorem exists_axis_net {δ : ℝ} (hδ : 0 < δ) : | |
| ∃ D : Finset ℂ, (∀ ν ∈ D, ‖ν‖ = 1) ∧ 1 ∈ D ∧ -1 ∈ D ∧ | |
| Complex.I ∈ D ∧ -Complex.I ∈ D ∧ | |
| (∀ ν : ℂ, ‖ν‖ = 1 → ∃ μ ∈ D, ‖ν-μ‖ < δ) := by | |
| classical | |
| obtain ⟨S,hS⟩ := (isCompact_sphere (0:ℂ) 1).elim_finite_subcover | |
| (fun ν : sphere (0:ℂ) 1 => ball (ν:ℂ) δ) (fun _ => isOpen_ball) (by | |
| intro ν hν | |
| exact mem_iUnion.mpr ⟨⟨ν,hν⟩,mem_ball_self hδ⟩) | |
| let D := S.image Subtype.val ∪ {1,-1,Complex.I,-Complex.I} | |
| refine ⟨D,?_,?_,?_,?_,?_,?_⟩ | |
| · intro ν hν | |
| rcases Finset.mem_union.mp hν with hν | hν | |
| · obtain ⟨μ,hμ,rfl⟩ := Finset.mem_image.mp hν | |
| simpa only [mem_sphere,dist_zero_right] using μ.property | |
| · simp only [Finset.mem_insert,Finset.mem_singleton] at hν | |
| rcases hν with rfl | rfl | rfl | rfl <;> simp | |
| · simp [D] | |
| · simp [D] | |
| · simp [D] | |
| · simp [D] | |
| · intro ν hν | |
| obtain ⟨μ,hμ,hνμ⟩ := mem_iUnion₂.mp (hS (by simpa only [mem_sphere,dist_zero_right] using hν)) | |
| refine ⟨μ,Finset.mem_union_left _ (Finset.mem_image.mpr ⟨μ,hμ,rfl⟩),?_⟩ | |
| simpa [dist_eq_norm] using hνμ | |
| def convexSupport (K : Set ℂ) (ν : ℂ) : ℝ := sSup ((fun z => inner ℝ ν z) '' K) | |
| theorem convexSupport_attained {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty) (ν : ℂ) : | |
| ∃ w ∈ K, inner ℝ ν w = convexSupport K ν := by | |
| exact (hK.image ((innerSL ℝ ν).continuous)).sSup_mem (hne.image _) | |
| theorem le_convexSupport {K : Set ℂ} (hK : IsCompact K) {z : ℂ} (hz : z ∈ K) (ν : ℂ) : | |
| inner ℝ ν z ≤ convexSupport K ν := by | |
| apply le_csSup (hK.image ((innerSL ℝ ν).continuous)).bddAbove | |
| exact mem_image_of_mem _ hz | |
| theorem convexSupport_le {K : Set ℂ} (hne : K.Nonempty) {M : ℝ} | |
| (hM : ∀ z ∈ K, ‖z‖ ≤ M) {ν : ℂ} (hν : ‖ν‖ = 1) : convexSupport K ν ≤ M := by | |
| apply csSup_le (hne.image _) | |
| rintro x ⟨z,hz,rfl⟩ | |
| exact (real_inner_le_norm ν z).trans (by simpa only [hν,one_mul] using hM z hz) | |
| theorem direction_net_distance {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty) | |
| (hconv : Convex ℝ K) {D : Finset ℂ} {δ M L : ℝ} (hδ : 0 < δ) | |
| (hM : ∀ w ∈ K, ‖w‖ ≤ M) | |
| (hnet : ∀ ν : ℂ, ‖ν‖ = 1 → ∃ μ ∈ D, ‖ν-μ‖ < δ) | |
| {z : ℂ} (hz : ‖z‖ ≤ L) | |
| (hs : ∀ μ ∈ D, inner ℝ μ z ≤ convexSupport K μ+δ) : | |
| ∃ w ∈ K, ‖z-w‖ ≤ δ*(1+L+M) := by | |
| obtain ⟨w,hw,hwmin⟩ := exists_norm_eq_iInf_of_complete_convex hne hK.isClosed.isComplete hconv z | |
| have hM0 : 0 ≤ M := (norm_nonneg w).trans (hM w hw) | |
| have hL0 : 0 ≤ L := (norm_nonneg z).trans hz | |
| refine ⟨w,hw,?_⟩ | |
| by_cases he : z = w | |
| · rw [he,sub_self,norm_zero] | |
| positivity | |
| have hn : 0 < ‖z-w‖ := norm_pos_iff.mpr (sub_ne_zero.mpr he) | |
| let ν := (‖z-w‖⁻¹:ℝ) • (z-w) | |
| have hν : ‖ν‖ = 1 := by simp [ν,ne_of_gt hn] | |
| obtain ⟨μ,hμ,hνμ⟩ := hnet ν hν | |
| obtain ⟨u,hu,hsu⟩ := convexSupport_attained hK hne μ | |
| have hmin : inner ℝ ν (u-w) ≤ 0 := by | |
| have hh := (norm_eq_iInf_iff_real_inner_le_zero hconv hw).mp hwmin u hu | |
| dsimp only [ν] | |
| rw [real_inner_smul_left] | |
| exact mul_nonpos_of_nonneg_of_nonpos (le_of_lt (inv_pos.mpr hn)) hh | |
| have hzw : inner ℝ ν (z-w) = ‖z-w‖ := by | |
| dsimp only [ν] | |
| rw [real_inner_smul_left,real_inner_self_eq_norm_sq] | |
| field_simp | |
| have hmain : ‖z-w‖ ≤ inner ℝ ν (z-u) := by | |
| have heq : z-w = (z-u)+(u-w) := by abel | |
| rw [← hzw,heq,inner_add_right] | |
| linarith | |
| have hμbound : inner ℝ μ (z-u) ≤ δ := by | |
| rw [inner_sub_right,hsu] | |
| linarith [hs μ hμ] | |
| have hnorm : ‖z-u‖ ≤ L+M := (norm_sub_le z u).trans (add_le_add hz (hM u hu)) | |
| have herror : inner ℝ (ν-μ) (z-u) ≤ δ*(L+M) := by | |
| calc | |
| inner ℝ (ν-μ) (z-u) ≤ ‖ν-μ‖*‖z-u‖ := real_inner_le_norm _ _ | |
| _ ≤ δ*(L+M) := mul_le_mul (le_of_lt hνμ) hnorm (norm_nonneg _) (le_of_lt hδ) | |
| have heq : inner ℝ ν (z-u) = inner ℝ μ (z-u)+inner ℝ (ν-μ) (z-u) := by | |
| rw [inner_sub_left]; ring | |
| rw [heq] at hmain | |
| nlinarith | |
| theorem axis_support_bound {K : Set ℂ} (hne : K.Nonempty) {M δ : ℝ} | |
| (hM : ∀ w ∈ K, ‖w‖ ≤ M) {D : Finset ℂ} | |
| (h1 : 1 ∈ D) (hm1 : -1 ∈ D) (hI : Complex.I ∈ D) (hmI : -Complex.I ∈ D) | |
| {z : ℂ} (hs : ∀ μ ∈ D, inner ℝ μ z ≤ convexSupport K μ+δ) : | |
| ‖z‖ ≤ 2*(M+δ) := by | |
| have hb (μ : ℂ) (hμ : μ ∈ D) (hμN : ‖μ‖ = 1) : inner ℝ μ z ≤ M+δ := | |
| (hs μ hμ).trans (add_le_add_left (convexSupport_le hne hM hμN) δ) | |
| have hr := hb 1 h1 (by simp) | |
| have hrr := hb (-1) hm1 (by simp) | |
| have hi := hb Complex.I hI (by simp) | |
| have hir := hb (-Complex.I) hmI (by simp) | |
| simp only [real_inner_eq_re_inner ℂ,RCLike.inner_apply] at hr hrr hi hir | |
| simp at hr hrr hi hir | |
| have ha : |z.re| ≤ M+δ := abs_le.mpr ⟨by linarith,hr⟩ | |
| have hb : |z.im| ≤ M+δ := abs_le.mpr ⟨by linarith,hi⟩ | |
| calc | |
| ‖z‖ ≤ |z.re|+|z.im| := Complex.norm_le_abs_re_add_abs_im z | |
| _ ≤ 2*(M+δ) := by linarith | |
| end DirectCrouzeix.Geometry | |
| noncomputable section | |
| open Set | |
| open scoped ComplexConjugate Matrix | |
| namespace DirectCrouzeix.Geometry | |
| end DirectCrouzeix.Geometry | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |