High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension

Abstract

Let A be a d by n matrix, d 0 with the property that, for any ρ ∼ ρN(δ), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 ≤ k ≤ ρd. This implies that P is centrally [ρd] -neighborly, and its skeleton Skel[ρd] (P) is combinatorially equivalent to Skel [ρd] (C). We display graphs of ρN. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: weak neighborliness and sectional neighborliness [9]: we study both. Weak (k,ε)-neighborliness asks if the &-faces are all simplicial and if the number of k-dimensional faces fk(P) > f k (C)(1 - ε). We characterize and compute the critical proportion ρW(δ) > 0 such that weak (k, ε) neighborliness holds at k significantly smaller than ρW · d and fails for k significantly larger than ρW · d. Sectional (k, ε)neighborliness asks whether all, except for a small fraction ε, of the k-dimensional intrinsic sections of P are k-dimensional cross polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ρρS(δ) > 0 guaranteeing this property for k/d ∼ ρ ≤ ρS(δ). We display graphs of ρS and ρW.