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 $\phi(T)=(z_1, \dots, z_n)$ 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 $\phi(T)$. 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\v{\i} in which they illuminate connections between regular triangulations and subdivisions of Newton polytopes, and generalized discriminants and determinants. The secondary polytope, $\Sigma (V)$, of an arbitrary finite point set $V \subset \mathbb R^d$, introduced by Gel'fand, Kapranov, and Zelevinski\v{\i}, is defined to be the convex hull of the GKZ-vectors of all triangulations of $V$. They showed the vertices of $\Sigma (V)$ 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.