File size: 2,953 Bytes
e4d9d2b | 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 | import OAI.Analysis.RCDHeat.InnerPath
import OAI.Analysis.RCDHeat.KantorovichRadius
namespace OAI
/-! # Compact contact completion by two quadratic transforms
The completion preserves every contact of the original compact potential.
It also keeps an explicit compact support radius for the right potential.
-/
open Set Metric
open scoped NNReal
noncomputable section
namespace RegularPointCoordinates
variable {X : Type*} [MetricSpace X] [ProperSpace X]
theorem compact_contact_completion (hgeo : IsGeodesicSpace X)
(p : X) {φ : X → ℝ} {L : ℝ≥0} (hφ : LipschitzWith L φ)
{R : ℝ} (hzero : ∀ x, R < dist p x → φ x = 0) :
∃ (ψ : X → ℝ) (r : ℝ), LipschitzWith L ψ ∧ HasCompactSupport ψ ∧
(∀ x, r < dist p x → ψ x = 0) ∧
LipschitzWith L (hopfLax (-ψ) 1) ∧ HasCompactSupport (hopfLax (-ψ) 1) ∧
(∀ x y, hopfLax (-ψ) 1 x + ψ y ≤ dist x y ^ 2 / 2) ∧
(∀ x, φ x ≤ hopfLax (-ψ) 1 x) ∧
∀ x y b, (∀ z, φ z + b ≤ dist z y ^ 2 / 2) →
φ x + b = dist x y ^ 2 / 2 →
hopfLax (-ψ) 1 x = φ x ∧ ψ y = b := by
let ψ := hopfLax (-φ) 1
have hpos : (0 : ℝ) < 1 := by norm_num
have hψ : LipschitzWith L ψ := hopfLax_lipschitz hφ.neg hgeo hpos
have hnegzero (x : X) (hx : R < dist p x) : (-φ) x = 0 := by
simp only [Pi.neg_apply,hzero x hx,neg_zero]
have hψs : HasCompactSupport ψ := hopfLax_hasCompactSupport hφ.neg p R hnegzero hpos
have hψzero (x : X) (hx : R+2*(L:ℝ) < dist p x) : ψ x = 0 := by
apply hopfLax_zero_outside hφ.neg p R hnegzero hpos
simpa only [mul_one] using hx
have hψnegzero (x : X) (hx : R+2*(L:ℝ) < dist p x) : (-ψ) x = 0 := by
simp only [Pi.neg_apply,hψzero x hx,neg_zero]
have hχ : LipschitzWith L (hopfLax (-ψ) 1) := hopfLax_lipschitz hψ.neg hgeo hpos
have hχs : HasCompactSupport (hopfLax (-ψ) 1) :=
hopfLax_hasCompactSupport hψ.neg p (R+2*(L:ℝ)) hψnegzero hpos
have had (x y : X) : φ x + ψ y ≤ dist x y ^ 2 / 2 := by
have hh := hopfLax_le_cost hφ.neg hpos y x
simp only [Pi.neg_apply,mul_one,dist_comm y x] at hh
dsimp only [ψ]
linarith
refine ⟨ψ,R+2*(L:ℝ),hψ,hψs,hψzero,hχ,hχs,?_,?_,?_⟩
· intro x y
have hh := hopfLax_le_cost hψ.neg hpos x y
simp only [Pi.neg_apply,mul_one] at hh
linarith
· intro x
apply le_csInf ⟨_,mem_range_self x⟩
rintro _ ⟨y,rfl⟩
have hh := had x y
simp only [Pi.neg_apply,mul_one]
linarith
· intro x y b hb hxy
have hby : b ≤ ψ y := by
apply le_csInf ⟨_,mem_range_self x⟩
rintro _ ⟨z,rfl⟩
have hh := hb z
simp only [Pi.neg_apply,mul_one,dist_comm y z]
linarith
have hyb : ψ y ≤ b := by linarith [had x y]
have he : ψ y = b := le_antisymm hyb hby
refine ⟨(hopfLax_of_contact hψ had (x := x) (y := y) ?_).1,he⟩
rwa [he]
end RegularPointCoordinates
end
end OAI
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