The polytope of all triangulations of a point configuration

Abstract

We study the convex hull P A of the 0-1 incidence vectors of all triangulations of a point connguration A. This was called the universal polytope in 4]. The aane span of P A is described in terms of the co-circuits of the oriented matroid of A. Its intersection with the positive orthant is a quasi-integral polytope Q A whose integral hull equals P A. We present the smallest example where Q A and P A diier. The duality theory for regular triangulations in 5] is extended to cover all triangulations. We discuss potential applications to enumeration and optimization problems regarding all triangulations.