On the polyhedral cones of convex and concave vectors

Abstract

Convex or concave sequences of n positive terms, viewed as vectors in n-space, constitute convex cones with 2n − 2 and n extreme rays, respectively. Explicit description is given of vectors spanning these extreme rays, as well as of non-singular linear transformations between the positive orthant and the simplicial cones formed by the positive concave vectors. The simplicial cones of monotone convex and concave vectors can be described similarly. In this note a sequence (vector) a = (a1, . . . , an), n ≥ 1, of real numbers is called positive if, for all 1 ≤ i ≤ n, 0 ≤ ai, increasing if for all 1 ≤ i 0, is sometimes based on proving the stronger property of concavity of (log a1, . . . , log an), called logconcavity [1, 2, 5, 9]. In turn, to prove that log-concavity is preserved in certain constructions of sequences, ordinary concavity of some coefficient sequences may be used [3]. These connections motivate our interest in the geometric description of the sets of vectors possessing one or another of the properties mentioned. Each of the sets of positive, negative, increasing, decreasing, convex, and concave vectors, and various intersections of these sets form closed cones (sets containing the null vector and closed under linear combinations with non-negative coefficients, called conic combinations). The cone of positive vectors is the positive orthant in R. The cone of positive increasing vectors was described implicitly by Lovasz ([4], p. 248, last equation) and by Marichal and Mathonet [6] in terms of its intersection with the unit hypercube (one of the n! simplices of the standard triangulation of the hypercube). In this note we describe the cones of positive concave and positive convex vectors by determining their extreme rays. When this cone is simplicial, we describe a standardized matrix realizing the transformation of the orthant to the cone in question. MSC (2010): primary 05D05, 05A20, 52A40; secondary 15B48, 15A39.