Weighted cone-volume measures of pseudo-cones

TL;DR

This paper introduces weighted cone-volume measures for pseudo-cones in Euclidean space, establishing conditions under which these measures are finite and characterizing measures that correspond to specific pseudo-cones.

Contribution

It defines weighted cone-volume measures for pseudo-cones, proves necessary conditions, and characterizes measures that arise from these pseudo-cones.

Findings

Weighted cone-volume measures can be finite with suitable weights.

Necessary and sufficient conditions for measures to be cone-volume measures of pseudo-cones.

Characterization of measures on the sphere corresponding to pseudo-cones.

Abstract

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.