Canonical Barriers on Convex Cones

Abstract

On the interior of a regular convex cone K in n-dimensional real space there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K, the universal barrier. It is invariant with respect to the unimodular automorphism subgroup of K and is compatible with the operation of taking product cones, but in general it does not behave well under duality. Here we introduce a barrier associated with the Cheng-Yau metric, the canonical barrier. It shares with the universal barrier the invariance, existence, and uniqueness properties and is compatible with the operation of taking product cones, but in addition is well behaved under duality. The canonical barrier can be characterized as the convex solution of the partial differential equation log det F” = 2F that tends to infinity as the argument tends to the boundary of K. Its barrier parameter does not exceed the dimension n of the cone. On homogeneous cones both barriers essentially coincide.