Centrally symmetric polytopes with many faces

TL;DR

This paper provides explicit constructions of high-dimensional centrally symmetric polytopes with many faces, demonstrating new combinatorial properties and face-spanning characteristics for such polytopes.

Contribution

The authors introduce novel explicit constructions of centrally symmetric polytopes with many faces, advancing understanding of their combinatorial and geometric properties.

Findings

Constructed a polytope with about (1.316)^d vertices where every non-antipodal pair forms an edge.

Developed polytopes with a large number of faces, covering a significant fraction of possible k-subsets.

Achieved polytopes with a high number of vertices and dimensions where most k-subsets span faces.

Abstract

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, second, for an integer k>1, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of P, and third, for an integer k>1 and a>0, we construct a centrally symmetric polytope Q with an arbitrary large number N of vertices and of dimension d=k^{1+o(1)} such that least (1 - k^{-a}){N choose k} k-subsets of the set of vertices span faces of Q.