| import Mathlib | |
| import OAI.Geometry.Borsuk.Projectors.Basic | |
| namespace OAI | |
| namespace BorsukNine | |
| open scoped RealInnerProductSpace ContDiff | |
| open scoped Manifold | |
| section HomogeneousLift | |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| {ι : Type*} | |
| noncomputable def homogeneousLift (f : V → ι → ℝ) (v : V) : ι → ℝ := | |
| fun i => ‖v‖ ^ 2 * f ((‖v‖⁻¹ : ℝ) • v) i | |
| @[simp] lemma homogeneousLift_zero (f : V → ι → ℝ) : homogeneousLift f 0 = 0 := by | |
| ext i | |
| simp [homogeneousLift] | |
| lemma homogeneousLift_unit (f : V → ι → ℝ) {v : V} (hv : ‖v‖ = 1) : | |
| homogeneousLift f v = f v := by | |
| ext i | |
| simp [homogeneousLift, hv] | |
| variable [Fintype ι] | |
| lemma Admissible.lift_nonneg {f : V → ι → ℝ} (hf : Admissible f) (v : V) (i : ι) : | |
| 0 ≤ homogeneousLift f v i := by | |
| by_cases hv : v = 0 | |
| · simp [hv] | |
| · exact mul_nonneg (sq_nonneg _) (hf.nonneg _ (norm_smul_inv_norm (𝕜 := ℝ) hv) i) | |
| lemma Admissible.lift_sum {f : V → ι → ℝ} (hf : Admissible f) (v : V) : | |
| ∑ i, homogeneousLift f v i = ‖v‖ ^ 2 := by | |
| classical | |
| by_cases hv : v = 0 | |
| · simp [hv] | |
| · simp only [homogeneousLift, ← Finset.mul_sum] | |
| have hn : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (norm_nonneg v))] | |
| exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hv) | |
| rw [hf.sum_one _ hn, mul_one] | |
| lemma Admissible.lift_abs_sum {f : V → ι → ℝ} (hf : Admissible f) (v : V) : | |
| ∑ i, |homogeneousLift f v i| = ‖v‖ ^ 2 := by | |
| simp only [abs_of_nonneg (hf.lift_nonneg v _), hf.lift_sum] | |
| lemma Admissible.lift_le {f : V → ι → ℝ} (hf : Admissible f) (v : V) (i : ι) : | |
| homogeneousLift f v i ≤ ‖v‖ ^ 2 := by | |
| classical | |
| rw [← hf.lift_sum v] | |
| exact Finset.single_le_sum (fun j _ => hf.lift_nonneg v j) (Finset.mem_univ i) | |
| lemma Admissible.lift_norm_le {f : V → ι → ℝ} (hf : Admissible f) (v : V) : | |
| ‖homogeneousLift f v‖ ≤ ‖v‖ ^ 2 := by | |
| rw [pi_norm_le_iff_of_nonneg (sq_nonneg _)] | |
| intro i | |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (hf.lift_nonneg v i)] using hf.lift_le v i | |
| lemma Admissible.lift_even {f : V → ι → ℝ} (hf : Admissible f) (v : V) : | |
| homogeneousLift f (-v) = homogeneousLift f v := by | |
| ext i | |
| simp [homogeneousLift, smul_neg, hf.antipodal] | |
| omit [Fintype ι] in | |
| lemma homogeneousLift_smul_pos (f : V → ι → ℝ) (v : V) {t : ℝ} (ht : 0 < t) : | |
| homogeneousLift f (t • v) = t ^ 2 • homogeneousLift f v := by | |
| ext i | |
| have he : (‖t • v‖⁻¹ : ℝ) • (t • v) = (‖v‖⁻¹ : ℝ) • v := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos ht, mul_inv_rev, smul_smul] | |
| congr 1 | |
| field_simp | |
| change ‖t • v‖ ^ 2 * f ((‖t • v‖⁻¹ : ℝ) • (t • v)) i = t ^ 2 * (‖v‖ ^ 2 * f ((‖v‖⁻¹ : ℝ) • v) i) | |
| rw [he, norm_smul, Real.norm_eq_abs, abs_of_pos ht] | |
| ring | |
| omit [Fintype ι] in | |
| lemma homogeneousLift_smul_nonneg (f : V → ι → ℝ) (v : V) {t : ℝ} (ht : 0 ≤ t) : | |
| homogeneousLift f (t • v) = t ^ 2 • homogeneousLift f v := by | |
| rcases ht.eq_or_lt with rfl | hp | |
| · simp | |
| · exact homogeneousLift_smul_pos f v hp | |
| lemma Admissible.lift_continuous {f : V → ι → ℝ} (hf : Admissible f) : | |
| Continuous (homogeneousLift f) := by | |
| apply continuous_iff_continuousAt.mpr | |
| intro v | |
| by_cases hv : v = 0 | |
| · subst v | |
| rw [ContinuousAt, homogeneousLift_zero] | |
| apply squeeze_zero_norm (fun x => hf.lift_norm_le x) | |
| have hc : Continuous (fun x : V => ‖x‖ ^ 2) := continuous_norm.pow 2 | |
| simpa using hc.tendsto (0 : V) | |
| · have hn : ContinuousAt (fun x : V => (‖x‖⁻¹ : ℝ) • x) v := | |
| (continuous_norm.continuousAt.inv₀ (norm_ne_zero_iff.mpr hv)).smul continuousAt_id | |
| apply continuousAt_pi.mpr | |
| intro i | |
| exact (continuous_norm.continuousAt.pow 2).mul ((hf.smooth i).continuous.continuousAt.comp hn) | |
| lemma Admissible.lift_orthogonal {f : V → ι → ℝ} (hf : Admissible f) | |
| {u v : V} (huv : ⟪u, v⟫ = 0) (i : ι) : | |
| homogeneousLift f u i * homogeneousLift f v i = 0 := by | |
| by_cases hu : u = 0 | |
| · simp [hu] | |
| by_cases hv : v = 0 | |
| · simp [hv] | |
| have hn : ⟪(‖u‖⁻¹ : ℝ) • u, (‖v‖⁻¹ : ℝ) • v⟫ = 0 := by | |
| simp [inner_smul_left, inner_smul_right, huv] | |
| have h := hf.orthogonal _ _ (norm_smul_inv_norm (𝕜 := ℝ) hu) (norm_smul_inv_norm (𝕜 := ℝ) hv) hn i | |
| have hnu := hf.nonneg _ (norm_smul_inv_norm (𝕜 := ℝ) hu) i | |
| have hnv := hf.nonneg _ (norm_smul_inv_norm (𝕜 := ℝ) hv) i | |
| have hz : f ((‖u‖⁻¹ : ℝ) • u) i = 0 ∨ f ((‖v‖⁻¹ : ℝ) • v) i = 0 := by | |
| by_contra hh | |
| push Not at hh | |
| exact h ⟨lt_of_le_of_ne hnu (Ne.symm hh.1), lt_of_le_of_ne hnv (Ne.symm hh.2)⟩ | |
| rcases hz with hz | hz <;> simp [homogeneousLift, hz] | |
| end HomogeneousLift | |
| open MeasureTheory Set Metric | |
| open scoped ENNReal Pointwise Matrix | |
| end BorsukNine | |
| end OAI | |