| 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 | |