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

Le résumé fourni par la source

We present explicit constructions of centrally symmetric 2-neighborly\nd-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally\nsymmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20\nk^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily\nlarge d and N, a centrally symmetric d-polytope with N vertices that has at\nleast (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where\ngamma_2=1/\\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another\napplication is a construction of a set of 3^{d/2 -1}-1 points in R^d every two\nof which are strictly antipodal as well as a construction of an n-point set\n(for an arbitrarily large n) in R^d with many pairs of strictly antipodal\npoints. The two latter results significantly improve the previous bounds by\nTalata, and Makai and Martini, respectively.\n

Ce résumé expose les affirmations des auteurs. BNTIC ne l’interprète pas comme une validation indépendante des résultats.