A weighted Minkowski theorem for pseudo-cones

TL;DR

This paper extends Minkowski's theorem to a new class of geometric objects called $C$-pseudo-cones, defining weighted measures and proving an existence theorem for these objects with prescribed measures.

Contribution

It introduces the concept of $C$-pseudo-cones, defines associated weighted surface area and covolume, and proves a Minkowski type existence theorem for these pseudo-cones.

Findings

Defined $C$-pseudo-cones with a fixed recession cone.

Introduced finite, weighted surface area and covolume measures.

Proved a Minkowski type existence theorem for $C$-pseudo-cones with given measures.

Abstract

A nonempty closed convex set in ${\mathbb R}^n$, not containing the origin, is called a pseudo-cone if with every $x$ it also contains $\lambda x$ for $x\ge 1$. We consider pseudo-cones with a given recession cone $C$, called $C$-pseudo-cones. The family of $C$-pseudo-cones can, with reasonable justification, be considered as a counterpart to the family of convex bodies containing the origin in the interior. For a $C$-pseudo-cone one can naturally define a surface area measure and a covolume. Since they are in general infinite, we introduce a weighting, leading to modified versions of surface area and covolume. These are finite and still homogeneous, though of different degrees. Our main result is a Minkowski type existence theorem for $C$-pseudo-cones with given weighted surface area measure.