New Bounds for the Integer Carathéodory Rank

Abstract

Given a rational pointed n-dimensional cone C, we study the integer Carathéodory rank CR(C) and its asymptotic form CR a (C), where we consider "most" integer vectors in the cone. The main result significantly improves the previously known upper bound for CR a (C). We also study bounds on CR(C) in terms of ∆, the maximal absolute n × n minor of the matrix given in an integral polyhedral representation of C. If ∆ ∈ {1, 2}, we show CR(C) = n, and prove upper bounds for simplicial cones, improving the best known upper bound on CR(C) for ∆ ≤ n.