| import Mathlib |
| import OAI.Geometry.Borsuk.SphereGeometry.HomogeneousLift |
| import OAI.Geometry.Borsuk.SphereGeometry.SphereMeasure |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace ContDiff |
| open scoped Manifold |
| open MeasureTheory Set Metric |
| open scoped ENNReal Pointwise Matrix |
| section IntegralLift |
| variable {ν ι : Type*} [Fintype ν] [Nonempty ν] [Fintype ι] |
| local notation "E" => EuclideanSpace ℝ ν |
|
|
| noncomputable def integralLift (f : E → ι → ℝ) (T : E →L[ℝ] E) : ι → ℝ := |
| fun i => (Fintype.card ν : ℝ) * |
| ∫ x : UnitSphere E, homogeneousLift f (T x) i ∂sphereProbability E |
|
|
| omit [Nonempty ν] in |
| lemma Admissible.integrand_continuous {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : |
| Continuous (fun x : UnitSphere E => homogeneousLift f (T x) i) := |
| (continuous_apply i).comp (hf.lift_continuous.comp (T.continuous.comp continuous_subtype_val)) |
|
|
| lemma Admissible.integrand_integrable {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : |
| Integrable (fun x : UnitSphere E => homogeneousLift f (T x) i) (sphereProbability E) := |
| (hf.integrand_continuous T i).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
|
|
| omit [Nonempty ν] in |
| lemma Admissible.integralLift_nonneg {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : 0 ≤ integralLift f T i := |
| mul_nonneg (Nat.cast_nonneg _) (integral_nonneg (fun _ => hf.lift_nonneg _ i)) |
|
|
| lemma Admissible.integralLift_sum {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) : ∑ i, integralLift f T i = |
| (Fintype.card ν : ℝ) * ∫ x : UnitSphere E, ‖T x‖ ^ 2 ∂sphereProbability E := by |
| classical |
| simp only [integralLift, ← Finset.mul_sum] |
| rw [← integral_finsetSum _ (fun i _ => hf.integrand_integrable T i)] |
| simp only [hf.lift_sum] |
|
|
| lemma Admissible.integralLift_pos_iff_sphere {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : |
| 0 < integralLift f T i ↔ ∃ x : UnitSphere E, 0 < homogeneousLift f (T x) i := by |
| have hc : (0 : ℝ) < Fintype.card ν := Nat.cast_pos.mpr Fintype.card_pos |
| rw [integralLift, mul_pos_iff_of_pos_left hc] |
| constructor |
| · intro h |
| by_contra hn |
| push Not at hn |
| have he : (fun x : UnitSphere E => homogeneousLift f (T x) i) = 0 := by |
| funext x |
| exact le_antisymm (hn x) (hf.lift_nonneg _ i) |
| rw [he] at h |
| change 0 < ∫ _ : UnitSphere E, (0 : ℝ) ∂sphereProbability E at h |
| simp only [integral_zero, lt_self_iff_false] at h |
| · rintro ⟨x, hx⟩ |
| exact integral_pos_of_integrable_nonneg_nonzero (hf.integrand_continuous T i) |
| (hf.integrand_integrable T i) (fun _ => hf.lift_nonneg _ i) hx.ne' |
| |
| lemma Admissible.integralLift_continuous {f : E → ι → ℝ} (hf : Admissible f) : |
| Continuous (integralLift f) := by |
| apply continuous_pi |
| intro i |
| apply continuous_const.mul |
| have hc : Continuous (fun p : (E →L[ℝ] E) × UnitSphere E => |
| homogeneousLift f (p.1 p.2) i) := |
| (continuous_apply i).comp (hf.lift_continuous.comp |
| (continuous_fst.clm_apply (continuous_subtype_val.comp continuous_snd))) |
| have hi := continuous_parametric_integral_of_continuous |
| (μ := sphereProbability E) |
| (f := fun (T : E →L[ℝ] E) (x : UnitSphere E) => homogeneousLift f (T x) i) |
| hc isCompact_univ |
| simpa only [setIntegral_univ] using hi |
| |
| lemma Admissible.integralLift_pos_iff {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : |
| 0 < integralLift f T i ↔ ∃ x : E, 0 < homogeneousLift f (T x) i := by |
| rw [hf.integralLift_pos_iff_sphere] |
| constructor |
| · rintro ⟨x, hx⟩ |
| exact ⟨x, hx⟩ |
| · rintro ⟨x, hx⟩ |
| have hx0 : x ≠ 0 := by |
| rintro rfl |
| simp at hx |
| have hn : 0 < ‖x‖⁻¹ := inv_pos.mpr (norm_pos_iff.mpr hx0) |
| refine ⟨⟨‖x‖⁻¹ • x, mem_sphere_zero_iff_norm.mpr (norm_smul_inv_norm (𝕜 := ℝ) hx0)⟩, ?_⟩ |
| simp only [map_smul, homogeneousLift_smul_pos f _ hn, Pi.smul_apply, smul_eq_mul] |
| exact mul_pos (sq_pos_of_pos hn) hx |
| |
| lemma Admissible.integralLift_support {f : E → ι → ℝ} (hf : Admissible f) |
| (T : E →L[ℝ] E) (i : ι) : |
| 0 < integralLift f T i ↔ ∃ v : E, |
| v ∈ LinearMap.range T.toLinearMap ∧ ‖v‖ = 1 ∧ 0 < f v i := by |
| rw [hf.integralLift_pos_iff] |
| constructor |
| · rintro ⟨x, hx⟩ |
| have hx0 : T x ≠ 0 := by |
| intro hz |
| simp only [hz, homogeneousLift_zero, Pi.zero_apply, lt_self_iff_false] at hx |
| refine ⟨‖T x‖⁻¹ • T x, ⟨‖T x‖⁻¹ • x, by simp⟩, |
| norm_smul_inv_norm (𝕜 := ℝ) hx0, ?_⟩ |
| exact (mul_pos_iff_of_pos_left (sq_pos_of_pos (norm_pos_iff.mpr hx0))).mp hx |
| · rintro ⟨v, ⟨x, rfl⟩, hv, hx⟩ |
| refine ⟨x, ?_⟩ |
| rw [homogeneousLift_unit f (show ‖T x‖ = 1 from hv)] |
| exact hx |
| |
| lemma Admissible.integralLift_orthogonal {f : E → ι → ℝ} (hf : Admissible f) |
| (T S : E →L[ℝ] E) (hTS : ∀ x y, ⟪T x, S y⟫ = 0) (i : ι) : |
| integralLift f T i * integralLift f S i = 0 := by |
| apply mul_eq_zero.mpr |
| by_contra hn |
| push Not at hn |
| have ht := lt_of_le_of_ne (hf.integralLift_nonneg T i) (Ne.symm hn.1) |
| have hs := lt_of_le_of_ne (hf.integralLift_nonneg S i) (Ne.symm hn.2) |
| obtain ⟨x, hx⟩ := (hf.integralLift_pos_iff T i).mp ht |
| obtain ⟨y, hy⟩ := (hf.integralLift_pos_iff S i).mp hs |
| have hp := hf.lift_orthogonal (hTS x y) i |
| exact (ne_of_gt (mul_pos hx hy)) hp |
| |
| omit [Nonempty ν] [Fintype ι] in |
| lemma integralLift_smul_nonneg (f : E → ι → ℝ) (T : E →L[ℝ] E) |
| {t : ℝ} (ht : 0 ≤ t) : integralLift f (t • T) = t ^ 2 • integralLift f T := by |
| ext i |
| simp only [integralLift, smul_apply, homogeneousLift_smul_nonneg f _ ht, |
| Pi.smul_apply, smul_eq_mul, integral_const_mul] |
| ring |
| |
| end IntegralLift |
| |
| end BorsukNine |
| |
| end OAI |
| |
