| import OAI.Analysis.CircleDomains.Topology.CompactPositiveFunctional |
| import Mathlib.Analysis.Convex.Cone.Extension |
| import Mathlib.Topology.ContinuousMap.Algebra |
|
|
| namespace OAI |
|
|
| /-! Hahn–Banach separation for a compact family of measures and a |
| sublinear majorant cost. -/ |
| open Set MeasureTheory |
| open scoped CompactlySupported |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma exists_compact_positive_functional_of_majorant_cost |
| {X : Type*} [TopologicalSpace X] [CompactSpace X] [Nonempty X] |
| (p : C_c(X, ℝ) → ℝ) |
| (phom : ∀ c : ℝ, 0 < c → ∀ f, p (c • f) = c * p f) |
| (padd : ∀ f g, p (f+g) ≤ p f + p g) |
| (pneg : ∀ f, (∀ x, 0 ≤ f x) → p (-f) = 0) |
| (c : ℝ) |
| (pconst : ∀ r : ℝ, r*c ≤ p |
| (r • (⟨1,HasCompactSupport.of_compactSpace (1 : C(X,ℝ))⟩ : C_c(X,ℝ)))) : |
| ∃ Λ : C_c(X,ℝ) →ₚ[ℝ] ℝ, |
| Λ ⟨1,HasCompactSupport.of_compactSpace (1 : C(X,ℝ))⟩ = c ∧ |
| ∀ f, Λ f ≤ p f := by |
| classical |
| let oneX : C_c(X,ℝ) := ⟨1,HasCompactSupport.of_compactSpace (1 : C(X,ℝ))⟩ |
| let V : Submodule ℝ C_c(X,ℝ) := Submodule.span ℝ {oneX} |
| let x₀ : X := Classical.choice inferInstance |
| let ev : C_c(X,ℝ) →ₗ[ℝ] ℝ := |
| { toFun := fun f => f x₀ |
| map_add' := fun _ _ => rfl |
| map_smul' := fun _ _ => rfl } |
| let l : V →ₗ[ℝ] ℝ := c • (ev.comp V.subtype) |
| have hl (f : V) : l f ≤ p f := by |
| obtain ⟨r,hr⟩ := Submodule.mem_span_singleton.mp f.property |
| change c * ev f.val ≤ p f.val |
| rw [← hr] |
| simpa [ev,oneX,mul_comm] using pconst r |
| obtain ⟨L,hL,hLp⟩ := exists_extension_of_le_sublinear |
| (⟨V,l⟩ : C_c(X,ℝ) →ₗ.[ℝ] ℝ) p phom padd hl |
| have hpos (f : C_c(X,ℝ)) (hf : 0 ≤ f) : 0 ≤ L f := by |
| have hn := hLp (-f) |
| rw [pneg f (fun x => hf x),map_neg] at hn |
| linarith |
| refine ⟨PositiveLinearMap.mk₀ L hpos,?_,hLp⟩ |
| have he := hL (⟨oneX,Submodule.mem_span_singleton_self oneX⟩ : V) |
| change L oneX = c |
| simpa [l,ev,oneX] using he |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
