Logarithmic Voronoi polytopes for discrete linear models

Abstract

We study logarithmic Voronoi cells for linear statistical models and partial linear models. The logarithmic Voronoi cells at points on such model are polytopes. To any $d$-dimensional linear model inside the probability simplex $\Delta_{n-1}$, we can associate an $n\times d$ matrix $B$. For interior points, we describe the vertices of these polytopes in terms of co-circuits of $B$. We also show that these polytopes are combinatorially isomorphic to the dual of a vector configuration with Gale diagram $B$. This means that logarithmic Voronoi cells at all interior points on a linear model have the same combinatorial type. We also describe logarithmic Voronoi cells at points on the boundary of the simplex. Finally, we study logarithmic Voronoi cells of partial linear models, where the points on the boundary of the model are especially of interest.