Constructions and complexity of secondary polytopes

Abstract

The secondary polytope Σ( A ) of a configuration A of n points in affine ( d − 1)-space is an ( n − d)-polytope whose vertices correspond to regular triangulations of conv( A ). In this article we present three constructions of Σ( A ) and apply them to study various geometric, combinatorial, and computational properties of secondary polytopes. The first construction is due to Gel'fand, Kapranov, and Zelevinsky, who used it to describe the face lattice of Σ( A ). We introduce the universal polytope u(A) ⊂ Λ d R n , a combinatorial object depending only on the oriented matroid of A . The secondary Σ( A ) can be obtained as the image of u(A) under a canonical linear map onto R n . The third construction is based upon Gale transforms or oriented matroid duality. It is used to analyze the complexity of computing Σ( A ) and to give bounds in terms of n and d for the number of faces of Σ( A ).