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 $$3^{d/2}\approx (1.73)^d$$ vertices and of centrally symmetric $$k$$-neighborly $$d$$-polytopes with about $$2^{{3d}/{20k^2 2^k}}$$ vertices. Using this result, we construct for a fixed $$k\ge 2$$ and arbitrarily large $$d$$ and $$N$$, a centrally symmetric $$d$$-polytope with $$N$$ vertices that has at least $$\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$$ faces of dimension $$k-1$$, where $$\gamma _2=1/\sqrt{3}\approx 0.58$$ and $$\gamma _k = 2^{-3/{20k^2 2^k}}$$ for $$k\ge 3$$. Another application is a construction of a set of $$3^{\lfloor d/2 -1\rfloor }-1$$ points in $$\mathbb R ^d$$ every two of which are strictly antipodal as well as a construction of an $$n$$-point set (for an arbitrarily large $$n$$) in $$\mathbb R ^d$$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively. [ABSTRACT FROM AUTHOR]