Stellar discriminants and equipartitions

Abstract

In this paper we study the following problem: given a geometric d-simplex Δ and the set S of n points in the interior of Δ, find a stellar subdivision of Δ, such that the interiors of all the d-simplices of that subdivision contain equally many points from S.We introduce the relevant for this problem notion of points being in general position, and give a precise geometric definition of the corresponding stellar discriminant. We show that if points of S are in general position, then such a stellar subdivision always exists, and present an algorithm to find its center using quadratic (in n) time. If the requirement of being in general position is dropped, this is no longer the case. We give an example where the minimal gap in the distribution of points in any stellar subdivision is linear in n.We then apply our result to a variety of contexts, specifically: fast barycentric embeddings of geometric simplicial complexes, equipartition problems in tropical geometry, and maintaining a balanced system of master sensors in a sensor network.