Journée transport optimal, équation de Monge-Ampère et applications

Canonical barriers on convex cones

par Prof. Roland HILDEBRAND (Université Grenoble-Alpes)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

The Calabi theorem states that for every regular convex cone K in R^n, the Monge-Ampère equation log det F” = 2F/n has a unique convex solution on the interior of K which tends to +infty on the boundary of K. It turns out that this solution is self-concordant and logarithmically homogeneous, and thus is a barrier which can be used for conic optimization. We consider different aspects of this barrier:

  • affine spheres as level surfaces
  • metrization of the interior of K by the Hessian metric F”
  • primal-dual symmetry
  • interpretation as a minimal Lagrangian submanifold in a certain para-Kähler space form
  • complex-analytic structure on 3-dimensional cones.
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