OAI/Analysis/AffineBernstein/GlobalFlatInverseLIntegral.lean

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1. theorem OAI.AffineBernstein.affineEpigraph_flat_chart_global_inverseArea_lintegral line 22

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   Global exact inverse-second-form area transport on the entire literal flat support chart. All metric and parametrization data are derived from the original strictly convex graph. Neither stationarity nor a measure-transport law is assumed.

   theorem affineEpigraph_flat_chart_global_inverseArea_lintegral {n : ℕ} {Ω : Set (Space n)}
       (hΩ : IsOpen Ω) (hcv : Convex ℝ Ω) {u : Space n → ℝ}
       (hu : ContDiffOn ℝ ∞ u Ω) (hp : ∀ x ∈ Ω, (hessian u x).PosDef)
       (a : Space n × ℝ) (L : (S × E) ≃L[ℝ] (Space n × ℝ))
       {B : Set S} (hB : IsOpen B)
       (hK : ∀ s ∈ B, IsCompact {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       (hzero : ∀ s ∈ B, (0 : E) ∈ interior {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       (q₀ : S × E) (bS : Module.Basis ι ℝ S) (bE : OrthonormalBasis (κ ⊕ Unit) ℝ E)
       (hd : inner ℝ q₀.2 (bE (Sum.inr ())) = 1)
       (J : Space n →L[ℝ] (S × E)) (hi : Function.Injective J)
       (hJ : ∀ v, inner ℝ (J v).2 (bE (Sum.inr ())) = 0)
       (e : Fin n ≃ ι ⊕ κ) (hJe : ∀ i, J (coordinateVector n i) = tubeTangent bS bE (e i))
   …

   Mentions: OAI.AffineBernstein.Space, OAI.AffineBernstein.hessian, OAI.AffineBernstein.affineEpigraphPullback, OAI.AffineBernstein.coordinateVector, OAI.AffineBernstein.tubeTangent