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.
Rolf Schneider
. WEIGHTED CONE-VOLUME MEASURES OF PSEUDO-CONES[J]. Acta mathematica scientia, Series B, 2025
, 45(1)
: 40
-51
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DOI: 10.1007/s10473-025-0104-z
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