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

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...

An Explicit Construction for Neighborly Centrally Symmetric Polytopes

A polytope P ⊂ R is centrally symmetric (cs, for short) if P = −P . A cs polytope P is k-neighborly if every set of k of its vertices, no two of which are antipodes, is the vertex set of a face of P . In their recent paper [7], Linial and Novik give probabilistic constructions for highly neighborly cs polytopes. Namely, based on probabilistic techniques due to Garnaev and Gluskin [4], they cons...

Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes

Centrally symmetric polytopes with many faces

We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a d-dimensional centrally symmetric polytope P with about 3d/4 ≈ (1.316)d vertices such that every pair of non-antipodal vertices of P spans an edge of P , (2) for an integer k ≥ 2, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitra...

How Neighborly Can a Centrally Symmetric Polytope Be?

We show that there exist k-neighborly centrally symmetric ddimensional polytopes with 2(n + d) vertices, where k(d, n) = Θ ( d 1 + log((d + n)/d) ) . We also show that this bound is tight.