On the Solution Sets of the Cone-Volume Map

Abstract: For a fixed set $U=(u_1,\ldots,u_m)$ of outer unit normals, we consider the cone-volume map which associates to a vector of support numbers $b\in\mathbb R^m_{&gt; 0}$ the cone-volume vector $γ\in\mathbb R<sup>m$ of the polytope [ P(U,b)={x\in\mathbb{R}^n:U^\top x\leq b}. ] We study its fibers $S(U,γ)$ over prescribed cone-volume vectors $γ$, with particular emphasis on finiteness and non-uniqueness. We give characterizations of the irreducibility of $U$ in terms of properties of the cone-volume map and its solution sets. In particular, for irreducible $U$, we show that $S(U,γ)$ is finite for all $γ$ outside a set of measure zero in the cone-volume cone, and we give an explicit upper bound for its cardinality. In the planar case, we prove that $S(U,γ)$ has semialgebraic dimension at most one for arbitrary $U$. Moreover, for every $U\in\mathcal{U}(2,m)$ with $m\geq 4$, there exists a cone-volume vector $γ$ for which $S(U,γ)$ contains at least two distinct elements. As a consequence, there exist two distinct smooth planar convex bodies with the same cone-volume measure. Finally, for $U$ in general position, we show that the number of solutions is odd for every cone-volume vector outside the exceptional set.