On the discrete logarithmic Minkowski problem in the plane

The paper characterizes the convex hull of the closure of the cone-volume set $C_{\text{cv}}(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U\subseteq {\mathbb{R}}^2,\text{pos}(U)={\mathbb{R}}^2$. 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 ${\mathbb{R}}^2$.