Journée transport optimal, équation de Monge-Ampère et applications
# Canonical barriers on convex cones

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

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

Description

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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