The Voronoi diagram of codimension-one varieties under polyhedral norms

Given a point $x$ in a real algebraic variety $X$, which points in space are closer to $x$ than to any other point in $X$? The answer to this question is precisely the so called Voronoi cell of $x$, and the set of all Voronoi cells of points in the variety is called the Voronoi diagram of $X$. Describing the Voronoi diagram of a variety is a fundamental problem in Metric Algebraic Geometry, and it has been previously studied for the Euclidean distance. In this talk we explore the description of the Voronoi diagram of algebraic varieties of codimension one when the distance arises from a polyhedral norm. We will exemplify these diagrams with varieties arising from Algebraic Statistics and optimal transport.