Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3 ≈ (1.73) vertices and of centrally symmetric k-neighborly d-polytopes with about 2 2 2 k vertices. Using this result, we construct for a fixed k ≥ 2 and arbitrarily large d and N , a centrally symmetric d-polytope with N vertices that has at least ( 1− k · (γk) ) ( N k ) faces of dimension k − 1, where γ2 = 1/ √ 3 ≈ 0.58 and γk = 2 2 2 k for k ≥ 3. Another application is a construction of a set of 3 − 1 points in R every two of which are strictly antipodal as well as a construction of an n-point set (for an arbitrarily large n) in R with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.