AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexOpenAEDifferentiable

import AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexAEDifferentiable
import Mathlib.Topology.Compactness.SigmaCompact

/-!
# Almost-everywhere differentiability on an open convex domain

`ConvexAEDifferentiable` proves the all-space finite-valued theorem.  Proper
convex functions arising from Rockafellar constructions are naturally finite
only on an effective domain, so the next reusable interface must keep the
finite region explicit.

For a real-valued convex function on an open convex set `s` in a
finite-dimensional normed space, we prove Frechet differentiability almost
everywhere **at points of `s`**.

The proof uses a sigma-compact exhaustion of the open subspace `s`.  Each
compact piece is enlarged, in the ambient locally compact space, to a compact
neighbourhood still contained in `s`.  Convexity gives local Lipschitz
regularity on `s`; compactness upgrades it to one Lipschitz constant on each
enlarged piece; Rademacher on its open interior gives ordinary Frechet
differentiability.  The interiors cover `s`.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace Analysis
namespace ConvexOpenAEDifferentiable

open MeasureTheory Metric Set Topology

noncomputable section

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
  {m : Measure E} [Measure.IsAddHaarMeasure m]

/-- A finite-valued convex function on an open convex domain of a
finite-dimensional real normed space is Frechet differentiable almost
everywhere at points of that domain. -/
theorem ae_differentiableAt_of_convexOn_isOpen
    {phi : E → ℝ} {s : Set E}
    (hs : IsOpen s) (hconv : ConvexOn ℝ s phi) :
    ∀ᵐ x ∂m, x ∈ s → DifferentiableAt ℝ phi x := by
  letI : LocallyCompactSpace s := hs.isLocallyClosed.locallyCompactSpace
  letI : SigmaCompactSpace s := inferInstance

  let K : ℕ → Set E := fun n =>
    Subtype.val '' SigmaCompactSpace.compactCovering s n
  have hKcompact : ∀ n, IsCompact (K n) := by
    intro n
    exact (SigmaCompactSpace.isCompact_compactCovering s n).image continuous_subtype_val
  have hKsub : ∀ n, K n ⊆ s := by
    intro n x hx
    rcases hx with ⟨y, _hy, rfl⟩
    exact y.property

  choose L hLcompact hKinterior hLsub using fun n =>
    exists_compact_between (hKcompact n) hs (hKsub n)

  have hcover : s ⊆ ⋃ n, interior (L n) := by
    intro x hx
    rcases SigmaCompactSpace.exists_mem_compactCovering (⟨x, hx⟩ : s) with ⟨n, hn⟩
    have hxK : x ∈ K n := by
      exact ⟨⟨x, hx⟩, hn, rfl⟩
    exact Set.mem_iUnion.2 ⟨n, hKinterior n hxK⟩

  have hlocal : LocallyLipschitzOn s phi :=
    hconv.locallyLipschitzOn hs
  have hcompactLip : ∀ n, ∃ C : ℝ≥0, LipschitzOnWith C phi (L n) := by
    intro n
    exact (hlocal.mono (hLsub n)).exists_lipschitzOnWith_of_compact (hLcompact n)
  choose C hLip using hcompactLip

  have hAe : ∀ n : ℕ, ∀ᵐ x ∂m,
      x ∈ interior (L n) → DifferentiableAt ℝ phi x := by
    intro n
    have hLipInterior : LipschitzOnWith (C n) phi (interior (L n)) :=
      (hLip n).mono interior_subset
    filter_upwards [hLipInterior.ae_differentiableWithinAt_of_mem] with x hx
    intro hxInt
    rcases hx hxInt with ⟨A, hA⟩
    exact ⟨A, (hasFDerivWithinAt_of_isOpen isOpen_interior hxInt).mp hA⟩

  filter_upwards [ae_all_iff.2 hAe] with x hxAll
  intro hx
  rcases Set.mem_iUnion.1 (hcover hx) with ⟨n, hxn⟩
  exact hxAll n hxn

end

end ConvexOpenAEDifferentiable
end Analysis
end TechnicalLemmas
end AutoSamplingTheory