On the discrete logarithmic Minkowski problem in the plane

TL;DR

This work addresses the discrete planar logarithmic Minkowski problem by analyzing the cone-volume set $C_{\tt cv}(U)$ associated with polygons whose outer normals lie in a finite set $U\subset\mathbb{S}^1$. The authors prove that the convex hull of the closure of $C_{\tt cv}(U)$ is a polytope with finitely many extreme points and provide both vertex (V) and half-space (H) representations, leading to new $U$-dependent necessary conditions for solvability in the plane. A key bridge is established with the subspace concentration polytope $P_{\tt scc}(U)$, showing $P_{\tt scc}(U) \subseteq \mathrm{cl}(\mathrm{conv}(C_{\tt cv}(U)))$, and the planar description is sharpened by a complete characterization of $\mathrm{cl}(\mathrm{conv}(C_{\tt cv}(U)))$ in terms of the sets $U_{\Delta}$, $U_{\square}$, and $U_{\square,u}$. The results illuminate the geometry of cone-volume vectors in the plane, clarifying when vertices arise from triangles versus $P_{\tt scc}$-vertices and detailing the trapezoid/parallelogram distinctions, with implications for algorithmic checks of necessary conditions for planar discrete logarithmic Minkowski problems.

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 $\R^2$.

Figures (5)

• Figure 1: The cone-volume set $C_{{\tt cv}}(U)$ is the union of the subsets represented in (A) and (B). Let $\gamma \in C_{\tt cv}(U)$. The $x-$axis corresponds to $\gamma_{1}$, the $y-$axis to $\gamma_{3}$ and the $z-$axis to $\gamma_2$. The corresponding vector in $C_{{\tt cv}}(U)$ is given via the formula $(\gamma_1,\gamma_2,\gamma_3,1-\gamma_1-\gamma_2-\gamma_3)$.
• Figure 2: The set $U$ drawn in the sphere with a open hemisphere $\omega \subset \mathbb{S}^1$ colored in red. The set $U$ allows us to construct trapezoids and triangles with the outer normals in $U$. Each $\gamma_i$ represents the cone-volume corresponding to the facet defined by $u_i$.
• Figure 3: The set $U$ with the open hemisphere $\omega \subset \mathbb{S}^1$ colored in red. This time we can only construct the parallelotope with the outer normals in $U$. Each $\gamma_i$ represents the cone-volume corresponding to the facet defined by $u_i$.
• Figure 4: Computing the hemisphere in the setting of Lemma \ref{['Lemma:StructureUSquare']}. The hemisphere $\omega \subset \mathbb{S}^1$ is colored red, and we rotated the set $U$ so that $u = e_1$. From the picture it is evident that $\mathop{\mathrm{pos}}\limits\{e_1,z,y'\} = \mathbb{R}^2$.
• Figure 5: The setting of Lemma \ref{['Lemma:DeltingAjacentFacetInceasreConeVolume']}, where we assume that $u_{i-1} = e_1$. On the left is a representation of $U$ drawn into the sphere. The hemisphere 'separating' $u_3$ and $u_m$ is colored in red. On the right we see a picture of the polytope $P$, which is limited by the black lines. The polytope $P'$ is the polytope $P$ togehter with the orange triangle. The red area corresponds to the parallelotope $A$ and the orange area is the triangle $\Delta$.

Theorems & Definitions (33)

• Theorem I
• Theorem II
• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Definition 2.2
• Proposition 2.3
• Lemma 2.4