| import Mathlib | |
| import OAI.Geometry.Borsuk.Support.FullSupport | |
| namespace OAI | |
| namespace BorsukNine | |
| open scoped RealInnerProductSpace ContDiff | |
| open scoped Manifold | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal Pointwise Matrix | |
| section SphereExtension | |
| open scoped MatrixOrder Matrix.Norms.L2Operator | |
| variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι] | |
| local notation "E" => EuclideanSpace ℝ ν | |
| lemma Admissible.signedExtension_eq_zero_iff {f : E → ι → ℝ} (hf : Admissible f) | |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) : signedExtension f A = 0 ↔ A = 0 := by | |
| constructor | |
| · intro hz | |
| have ht : Matrix.trace (CFC.abs A) = 0 := by | |
| rw [← hf.signedExtension_abs_sum hA, hz] | |
| simp | |
| have habs : CFC.abs A = 0 := matrix_psd_trace_zero (CFC.abs_nonneg A) ht | |
| have hsq := CFC.abs_mul_abs A | |
| rw [habs, zero_mul] at hsq | |
| apply Matrix.conjTranspose_mul_self_eq_zero.mp | |
| simpa only [Matrix.star_eq_conjTranspose] using hsq.symm | |
| · rintro rfl | |
| simp [signedExtension] | |
| noncomputable def normalizedSignedExtension (f : E → ι → ℝ) (A : Matrix ν ν ℝ) : | |
| EuclideanSpace ℝ ι := | |
| ‖WithLp.toLp 2 (signedExtension f A)‖⁻¹ • WithLp.toLp 2 (signedExtension f A) | |
| lemma Admissible.normalizedSignedExtension_norm {f : E → ι → ℝ} (hf : Admissible f) | |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) (hneq : A ≠ 0) : | |
| ‖normalizedSignedExtension f A‖ = 1 := by | |
| have hn : WithLp.toLp 2 (signedExtension f A) ≠ 0 := by | |
| intro hz | |
| have hs : signedExtension f A = 0 := WithLp.toLp_injective 2 hz | |
| exact hneq ((hf.signedExtension_eq_zero_iff hA).mp hs) | |
| simp [normalizedSignedExtension, norm_smul, norm_inv, hn] | |
| omit [Nonempty ν] in | |
| lemma normalizedSignedExtension_odd (f : E → ι → ℝ) (A : Matrix ν ν ℝ) : | |
| normalizedSignedExtension f (-A) = -normalizedSignedExtension f A := by | |
| simp [normalizedSignedExtension, signedExtension_odd, WithLp.toLp_neg] | |
| lemma Admissible.normalizedSignedExtension_continuousOn {f : E → ι → ℝ} (hf : Admissible f) : | |
| ContinuousOn (normalizedSignedExtension f) {A : Matrix ν ν ℝ | IsSelfAdjoint A ∧ A ≠ 0} := by | |
| have hc : ContinuousOn (fun A : Matrix ν ν ℝ => WithLp.toLp 2 (signedExtension f A)) | |
| {A | IsSelfAdjoint A ∧ A ≠ 0} := | |
| (PiLp.continuous_toLp 2 _).comp_continuousOn | |
| (hf.signedExtension_continuousOn.mono (fun _ h => h.1)) | |
| apply ContinuousOn.smul (ContinuousOn.inv₀ hc.norm _) hc | |
| intro A hA | |
| rw [norm_ne_zero_iff] | |
| intro hh | |
| have hs : signedExtension f A = 0 := WithLp.toLp_injective 2 hh | |
| exact hA.2 ((hf.signedExtension_eq_zero_iff hA.1).mp hs) | |
| end SphereExtension | |
| end BorsukNine | |
| end OAI | |