| import OAI.Geometry.RuledCovering.Main |
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| namespace OAI |
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| noncomputable section |
| open Set |
| namespace RuledApproximation |
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| theorem compact_spectral_envelope {X : Type*} [TopologicalSpace X] |
| {D : Set X} (hD : IsCompact D) {A : X → Plane →L[ℝ] Plane} |
| (hA : ContinuousOn A D) |
| (htr : ∀ p ∈ D, LinearMap.trace ℝ Plane (A p).toLinearMap = 0) |
| (hdet : ∀ p ∈ D, LinearMap.det (A p).toLinearMap < 0) : |
| ∃ m B : ℝ, 0 < m ∧ 0 < B ∧ ∀ p ∈ D, |
| m ≤ Real.sqrt (-LinearMap.det (A p).toLinearMap) ∧ |
| Real.sqrt (-LinearMap.det (A p).toLinearMap) ≤ B ∧ ‖A p‖ ≤ B := by |
| by_cases hne : D.Nonempty |
| · have hlam : ContinuousOn (fun p => Real.sqrt (-LinearMap.det (A p).toLinearMap)) D := |
| Real.continuous_sqrt.comp_continuousOn |
| (ContinuousLinearMap.continuous_det.comp_continuousOn hA).neg |
| obtain ⟨p, hp, hmin⟩ := hD.exists_isMinOn hne hlam |
| obtain ⟨B, hB⟩ := hD.exists_bound_of_continuousOn hA |
| refine ⟨Real.sqrt (-LinearMap.det (A p).toLinearMap), max 1 B, |
| Real.sqrt_pos.mpr (neg_pos.mpr (hdet p hp)), lt_of_lt_of_le zero_lt_one (le_max_left _ _), ?_⟩ |
| intro q hq |
| refine ⟨hmin hq, ?_, (hB q hq).trans (le_max_right _ _)⟩ |
| exact (eigenvalue_le_norm (A q) (Real.sqrt_nonneg _) |
| (plane_hyperbolic_eigenvalues (A q).toLinearMap (htr q hq) (hdet q hq)).1).trans |
| ((hB q hq).trans (le_max_right _ _)) |
| · refine ⟨1, 1, by norm_num, by norm_num, ?_⟩ |
| intro p hp |
| exact (hne ⟨p, hp⟩).elim |
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| end RuledApproximation |
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| end |
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| end OAI |
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