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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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.MatrixKernel
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace ContDiff
open scoped Manifold
open MeasureTheory Set Metric
open scoped ENNReal Pointwise Matrix
section RankOne
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι]
local notation "E" => EuclideanSpace ℝ ν
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
omit [DecidableEq ν] [Nonempty ν] in
lemma Admissible.lift_smul {f : E → ι → ℝ} (hf : Admissible f) (v : E) (t : ℝ) :
homogeneousLift f (t • v) = t ^ 2 • homogeneousLift f v := by
rcases le_total 0 t with ht | ht
· exact homogeneousLift_smul_nonneg f v ht
· have h := homogeneousLift_smul_nonneg f v (neg_nonneg.mpr ht)
rw [neg_smul, hf.lift_even, neg_sq] at h
exact h
noncomputable def rankOne (u : E) : Matrix ν ν ℝ := fun i j => u i * u j
omit [Nonempty ν] in
lemma rankOne_action (u x : E) : op (rankOne u) x = ⟪u, x⟫ • u := by
ext i
change ∑ j, (u i * u j) * x j = ⟪u, x⟫ * u i
simp_rw [mul_assoc]
rw [← Finset.mul_sum]
rw [PiLp.inner_apply]
simp only [RCLike.inner_apply, conj_trivial]
simp only [mul_comm]
omit [DecidableEq ν] [Nonempty ν] in
lemma rankOne_nonneg (u : E) : 0 ≤ rankOne u := by
apply Matrix.nonneg_iff_posSemidef.mpr
apply Matrix.posSemidef_iff_dotProduct_mulVec.mpr
constructor
· ext i j
simp [rankOne, Matrix.conjTranspose_apply, mul_comm]
· intro x
have he : star x ⬝ᵥ (rankOne u) *ᵥ x = (∑ i, u i * x i) ^ 2 := by
simp only [dotProduct, Matrix.mulVec, rankOne, Pi.star_apply, star_trivial]
simp_rw [mul_assoc, ← Finset.mul_sum]
simp_rw [← mul_assoc, mul_comm (x _) (u _)]
rw [← Finset.sum_mul, pow_two]
rw [he]
exact sq_nonneg _
omit [DecidableEq ν] [Nonempty ν] in
lemma rankOne_trace (u : E) : Matrix.trace (rankOne u) = ‖u‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply]
simp [Matrix.trace, Matrix.diag, rankOne, pow_two]
omit [Nonempty ν] in
lemma rankOne_mul_self (u : E) (hu : ‖u‖ = 1) : rankOne u * rankOne u = rankOne u := by
apply (op).injective
rw [map_mul]
apply ContinuousLinearMap.ext
intro x
change op (rankOne u) (op (rankOne u) x) = op (rankOne u) x
simp only [rankOne_action, inner_smul_right, real_inner_self_eq_norm_sq, hu, one_pow, mul_one]
omit [Nonempty ν] in
lemma rankOne_sqrt (u : E) (hu : ‖u‖ = 1) : CFC.sqrt (rankOne u) = rankOne u :=
(CFC.sqrt_eq_iff _ _ (rankOne_nonneg u) (rankOne_nonneg u)).mpr
(rankOne_mul_self u hu)
lemma Admissible.positiveExtension_rankOne {f : E → ι → ℝ} (hf : Admissible f)
(u : E) (hu : ‖u‖ = 1) : positiveExtension f (rankOne u) = f u := by
have hcoord (x : UnitSphere E) (i : ι) :
homogeneousLift f (op (rankOne u) x) i = ⟪u, (x : E)⟫ ^ 2 * f u i := by
rw [rankOne_action, hf.lift_smul, homogeneousLift_unit f hu]
rfl
have hm : (Fintype.card ν : ℝ) *
∫ x : UnitSphere E, ⟪u, (x : E)⟫ ^ 2 ∂sphereProbability E = 1 := by
have hs := hf.positiveExtension_sum (rankOne_nonneg u)
rw [rankOne_trace, hu, one_pow] at hs
simp only [positiveExtension, rankOne_sqrt u hu, integralLift] at hs
simp_rw [hcoord] at hs
simp only [integral_mul_const, ← Finset.mul_sum] at hs
rw [hf.sum_one u hu, mul_one] at hs
exact hs
ext i
simp only [positiveExtension, rankOne_sqrt u hu, integralLift]
simp_rw [hcoord]
rw [integral_mul_const, ← mul_assoc, hm, one_mul]
end RankOne
end BorsukNine
end OAI
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