OAI/Analysis/AffineBernstein/ActualTubeArea.lean

GitHub source file

3 declarations, 0 contain sorry, 0 declared as axioms

1. lemma OAI.AffineBernstein.affineEpigraph_support_jets line 21
2. theorem OAI.AffineBernstein.affineEpigraph_tube_positive line 56
3. theorem OAI.AffineBernstein.affineEpigraph_tube_area line 86

   Source comment

   Source tube-area, for the literal supremum/Gauss parametrization of the original affine epigraph under an arbitrary ambient affine isomorphism.

   theorem affineEpigraph_tube_area {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})
       {s : S} (hs : s ∈ B) {e : E}
       (bS : Module.Basis ι ℝ S) (bE : OrthonormalBasis (κ ⊕ Unit) ℝ E)
       (he : bE (Sum.inr ()) = 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