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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
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