Abstract
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 arbitrarily large number N of vertices such that for some 0 < δ k < 1 at least (1 − (δ k )d)( N k ) k-subsets of the set of vertices span faces of P, and (3) for an integer k ≥ 2 and α > 0, we construct a centrally symmetric polytope Q with an arbitrarily large number of vertices N and of dimension d = k 1+o(1) such that at least \((1 - k^{ - \alpha } )(_k^N )\) k-subsets of the set of vertices span faces of Q.
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The research of the first and second authors was partially supported by NSF Grant DMS-0856640.
The research of the third author was partially supported by NSF Grants DMS-0801152 and DMS-1069298.
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Barvinok, A., Lee, S.J. & Novik, I. Centrally symmetric polytopes with many faces. Isr. J. Math. 195, 457–472 (2013). https://doi.org/10.1007/s11856-012-0107-z
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DOI: https://doi.org/10.1007/s11856-012-0107-z