Special Vinberg cones and their applications

The talk is based on joint works with Vicete Cortes, Andrea Spiro and Alessio Marrani.
A short survey of the Vinberg theory of convex cones (including its informational geometric interpretation) and homogeneous convex cones will be presented. Then we concentrate on the theory of rank 3 special Vinberg cones, associated to metric Clifford $Cl({\mathbb R}^n)$ modules.
A generalization of the theory to the indefinite special Vinberg cones, associated to indefinite metric Clifford $Cl({\mathbb R}^{p,q})$ modules is indicated. An application of special Vinberg cones to $N=2 , \, d=5,4,3$ Supergravity will be considered.
We will discuss also applications of theory of homogeneous convex cones to convex programming, information geometry and Frobenius manifolds.