Explicit constructions of centrally symmetric k-neighborly polytopes and large strictly antipodal sets

TL;DR

This paper provides explicit constructions of symmetric polytopes with many faces and vertices, and large antipodal point sets, improving previous bounds and advancing geometric combinatorics.

Contribution

It introduces new explicit constructions of symmetric k-neighborly polytopes and large antipodal point sets, with significantly improved bounds over prior work.

Findings

Constructed centrally symmetric 2-neighborly polytopes with about (1.73)^d vertices.

Developed k-neighborly polytopes with exponentially many vertices in dimension d.

Created large antipodal point sets with many pairs of strictly antipodal points.

Abstract

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily large d and N, a centrally symmetric d-polytope with N vertices that has at least (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where gamma_2=1/\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another application is a construction of a set of 3^{d/2 -1}-1 points in R^d every two of which are strictly antipodal as well as a construction of an n-point set (for an arbitrarily large n) in R^d with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.