Unimodality and log-concavity off-vectors for cyclic and ordinary polytopes

Abstract

Ordinary polytopes are known as a non-simplicial generalization of the cyclic polytopes. The face vectors of ordinary polytopes are shown to be log-concave.

Log-concavity of rows of Pascal type triangles

“…Kurtz's method works to establish also the log-concavity of rows of certain convolution triangles, which was used in particular to prove log-concavity of f -vectors of ordinary polytopes (see [16], where convolution triangles have eventually decreasing, not necessarily log-concave initial side sequences). Proposition 3 works also to establish log-concavity of some triangular arrays where a row does not fully determine the next row (classical and generalized Delannoy triangles).…”

“…. ), as appearing implicitly in Hoggar [9], provides a simple approach to verify the log-concavity of f -vectors of cyclic polytopes, the first proof of this log-concavity property for the larger class of ordinary polytopes was based on an argument amounting to an instance of Kurtz's method [16].…”

“…Stirling numbers [18], f -vectors of polytopes). Kurtz's construction involves row to next row recursion using weighted summation of the two entries above a given entry [12], a simpler instance of which was used to study f -vectors of cyclic polytopes [16]. Brenti's more recent weighted summation formula is more complex than Kurtz's, in that it involves more than a single preceding row, but it also has different assumptions about the weights [5].…”

On the polyhedral cones of convex and concave vectors

“…The question of unimodality of the members of a class of sequences naturally arising in combinatorics can be difficult (e.g. Whitney numbers [2,7] and face vectors of certain classes of polytopes [8,5]). Proof of unimodality of a = a 1 , .…”

“…. , log a n (log-concavity) [1,2,9,5]. In turn, to prove that log-concavity is preserved in certain constructions of sequences, ordinary concavity of some coefficient sequences may be used [3].…”