Polytopes with Special Simplices

TL;DR

This paper classifies polytopes with special simplices, introduces the concepts of meek and wild polytopes, and explores their geometric and combinatorial properties, including construction methods and bounds on their face vectors.

Contribution

It provides a complete combinatorial classification of meek polytopes with special simplices and describes how wild polytopes relate to meek ones through hyperplane intersections.

Findings

Every wild polytope with a special simplex can be constructed from a meek polytope.

The f-vector of wild polytopes is component-wise bounded by that of a specific meek polytope.

The n-cube's basis polytope is a zonotope formed by Minkowski sum of an (n-1)-cube and a vector.

Abstract

For a polytope P a simplex S with vertex set V(S) is called a special simplex if every facet of P contains all but exactly one vertex of S. For such polytopes P with face complex F(P) containing a special simplex the subcomplex F(P) / V(S) of all faces not containing vertices of S is the boundary of a polytope Q - the basis polytope of P. If additionally the dimension of the affine basis space of F(P) / V(S) equals dim(Q), we call P meek; otherwise we call P wild. We give a full combinatorial classification and techniques for geometric construction of the class of meek polytopes with special simplices. We show that every wild polytope P' with special simplex can be constructed out of a particular meek one P by intersecting P with particular hyperplanes. It is non-trivial to find all these hyperplanes for an arbitrary basis polytope; we give an exact description for 2-basis polytopes. Furthermore we show that the f-vector of each wild polytope with special simplex is component wise bounded above by the f-vector of a particular meek one which can be computed explicitly. Finally, we discuss the n-cube as a non-trivial example of a wild polytope with special simplex and prove that its basis polytope is the zonotope given by the Minkowski sum of the (n-1)-cube and the vector (1,...,1). Polytopes with special simplex have applications on Ehrhart theory, toric rings and were just used by Francisco Santos to construct a counter-example disproving the Hirsch conjecture.