Recovering Lexicographic Triangulations

Abstract

Given a finite set $V=\{v^1, \dots, v^n\} \subset \mathbb R^d$ with dim conv $(V)=d$, a triangulation $T$ of $V$ is a collection of distinct subsets $\{T_1, \dots, T_m\}$ where $T_i \subseteq V$ is the vertex set of a $d$-simplex, $\mathrm{conv} (V)=\bigcup_{i=1}^m \mathrm{conv} (T_i)$, and $T_i \cap T_j$ is a common (possibly empty) face of both $T_i$ and $T_j$. Associated with each triangulation $T$ of $V$ is the GKZ-vector $ϕ(T)=(z_1, \dots, z_n)$ where $z_i$ is the sum of the volumes of all $d$-simplices of $T$ having $v^i \in V$ as a vertex. It is clear that given $V$ and a triangulation $T$ we can find $ϕ(T)$. The focus of this paper is recovering a lexicographic triangulation from its GKZ-vector. The motivation for studying triangulations and their GKZ-vectors arises from the work of Gel'fand, Kapranov, and Zelevinski\vı in which they illuminate connections between regular triangulations and subdivisions of Newton polytopes, and generalized discriminants and determinants. The secondary polytope, $Σ(V)$, of an arbitrary finite point set $V \subset \mathbb R^d$, introduced by Gel'fand, Kapranov, and Zelevinski\vı, is defined to be the convex hull of the GKZ-vectors of all triangulations of $V$. They showed the vertices of $Σ(V)$ are in one-to-one correspondence with the regular triangulations of $V$. Since the GKZ-vector of a regular triangulation is uniquely associated with that triangulation, a natural question is how that triangulation can be recovered from its vector. We answer this question in the case that the associated triangulation is lexicographic.