Weighted cone-volume measures of pseudo-cones

A pseudo-cone in ${\mathbb{R}}^n$ is a nonempty closed convex set $K$ not containing the origin and such that $\lambda K\subseteq K$ for all $\lambda \ge 1$. It is called a $C$-pseudo-cone if $C$ is its recession cone, where $C$ is a pointed closed convex cone with interior points. The cone-volume measure of a pseudo-cone can be defined similarly as for convex bodies, but it may be infinite. After proving a necessary condition for cone-volume measures of $C$-pseudo-cones, we introduce suitable weights for cone-volume measures, yielding finite measures. Then we provide a necessary and sufficient condition for a Borel measure on the unit sphere to be the weighted cone-volume measure of some $C$-pseudo-cone.