Canonical barriers on convex cones

par Prof. Roland HILDEBRAND (Université Grenoble-Alpes)

mercredi 7 déc. 2016, 14:40 → 15:30 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

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.