Weighted cone-volume measures of pseudo-cones

A pseudo-cone in Rⁿ is a nonempty closed convex set K not containing the origin and such that K K for all 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.