OAI/Analysis/AffineBernstein/SphericalAreaContinuity.lean

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2 declarations, 0 contain sorry, 0 declared as axioms

1. lemma OAI.AffineBernstein.continuousAt_sphericalArea line 20
2. theorem OAI.AffineBernstein.affineEpigraph_sphere_area_two_flat_lintegral line 44

   Source comment

   The actual full-sphere affine area is the sum of the two actual flat chart areas. Both flat charts are complete; their omitted equator has actual sphere measure zero.

   theorem affineEpigraph_sphere_area_two_flat_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))
       {D : Set S} (hD : IsOpen D)
       (hK : ∀ s ∈ D, IsCompact {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       (hzero : ∀ s ∈ D, (0:E) ∈ interior {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       {s : S} (hs : s ∈ D) (bS : Module.Basis ι ℝ S)
       (bF : OrthonormalBasis κ ℝ F) (bRef : OrthonormalBasis (κ ⊕ Unit) ℝ E)
       (f : WithLp 2 (F × ℝ) ≃ₗᵢ[ℝ] E)  :
       let H := fun q : S × E => homogeneousSupport {y | (q.1,y) ∈ affineEpigraphPullback Ω u a L} q.2
   …

   Mentions: OAI.AffineBernstein.Space, OAI.AffineBernstein.hessian, OAI.AffineBernstein.affineEpigraphPullback, OAI.AffineBernstein.homogeneousSupport