If we exclude simplices, this is the maximum possible k: in fact, every polytope that is k-neighborly for some k ≥ 1 + ⌊d⁄2⌋ is a simplex.
More generally, in any k-neighborly polytope, all faces of dimension less than k are simplices.
The cyclic polytopes formed as the convex hulls of finite sets of points on the moment curve (t, t2, …, td) in d-dimensional space are automatically neighborly.
[3] The convex hull of a set of random points, drawn from a Gaussian distribution with the number of points proportional to the dimension, is with high probability k-neighborly for a value k that is also proportional to the dimension.
A generalized version of the happy ending problem applies to higher-dimensional point sets, and implies that for every dimension d and every n > d there exists a number m(d,n) with the property that every m points in general position in d-dimensional space contain a subset of n points that form the vertices of a neighborly polytope.