High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension

Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in R. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax; [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly-projected cross-polytopes in the proportionaldimensional case where d ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R. We derive ρN (δ) > 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 bρdc-neighborly, and its skeleton Skelbρdc(P ) is combinatorially equivalent to Skelbρdc(C). We display graphs of ρN . Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: facial neighborliness and sectional neighborliness [9]; we study both. The weakest, (k, )-facial neighborliness, asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P ) ≥ fk(C)(1 − ). We characterize and compute the critical proportion ρF (δ) > 0 at which phase transition occurs in k/d. The other, (k, )sectional neighborliness, asks whether all, except for a small fraction , 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 ρS(δ) > 0 guaranteeing this property for k/d ∼ ρ < ρS(δ). We display graphs of ρS and ρF .