File size: 2,714 Bytes
8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
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