The canonical potential of a convex cone

English

Séminaire Modèles et Algorithmes Déterministes: CASYS

8/11/2012 - 09:45 Mr Daniel Fox (Universidad Politecnica de Madrid) Salle 1 - Tour IRMA

On a proper open convex cone there is a unique smooth convex barrier function equal to half the logarithm of the determinant of its Hessian and such that its Hessian is a complete Riemannian metric on the interior of the cone. Its level sets are hyperbolic affine spheres foliating the interior of the cone and asymptotic to its boundary. The talk aims to explain some aspects of the geometry associated to this function, in particular how the theorem, due independently to R. Hildebrand, that it is a self-concordant normal barrier with parameter equal to the dimension of the ambient space can be understood as a manifestation of the nonpositivity of the Ricci curvature of the associated Hessian metric. Some comments will be made about what happens in the nonconvex setting.