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

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 3d/20k 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 2 · (γ k ) d N k faces of dimension k − 1, where γ 2 = 1/ √ 3 ≈ 0.58 and γ k = 2 −3/20k 2 2 k for k ≥ 3. 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.

Combinatorial Distance Geometry in Normed Spaces

“…Talata improved this by a construction (described in [31, Lemma 9.11.2]) to A (d) Ω(3 d/3 ), and announced that A (d) Ω(5 d/4 ) (see [31, p. 271]). Subsequently, Barvinok, Lee, and Novik [15] found another construction that shows A (d) Ω(3 d/2 ). This is currently the best-known bound for e(X d ) for strictly convex spaces.…”

“…Makai and Martini [122] showed that a (3) 3, and conjectured that equality holds. Barvinok, Lee, and Novik [15] showed that a (d) Ω(3 d/2 ). The best upper bound known is the almost trivial a (d) 2 d − 1.…”

Acute Sets of Exponentially Optimal Size

“…The best earlier lower bound was 3 ⌊d/2⌋−1 − 1 due to Barvinok, Lee and Novik [2]. Note that the same upper bound (2 d − 1) holds for strictly antipodal sets as well [3].…”

Moment curves and cyclic symmetry for positive Grassmannians

“…Motivated by establishing an upper bound theorem for centrally symmetric polytopes (which is an open problem, even for 4‐dimensional polytopes), Barvinok and Novik employed symmetric moment curves to define a centrally symmetric analogue of cyclic polytopes, called bicyclic polytopes . They used bicyclic polytopes to give lower bounds on the numbers of faces of centrally symmetric polytopes, which they and Lee improved with polytopes constructed from variants of the symmetric moment curve. We refer to the survey of Novik for a detailed discussion of this topic.…”