On the discrete logarithmic Minkowski problem in the plane

TL;DR

This paper investigates the structure of cone-volume sets for polygons in the plane with normals in a finite set, providing new conditions for solving the logarithmic Minkowski problem in two dimensions.

Contribution

It characterizes the convex hull of cone-volume sets for polygons with normals in a finite set and derives new existence conditions for the logarithmic Minkowski problem in the plane.

Findings

Convex hull of cone-volume sets has finitely many extreme points.

Provided vertex and half-space representations of the convex hull.

Derived new necessary conditions for solutions to the logarithmic Minkowski problem.

Abstract

The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$.