The Schur Cone and the Cone of Log Concavity

Let {h1, h2, . . . } be a set of algebraically independent variables. We ask which vectors are extreme in the cone generated by hihj − hi+1hj−1 (i ≥ j > 0) and hi (i > 0). We call this cone the cone of log concavity. More generally, we ask which vectors are extreme in the cone generated by Schur functions of partitions with k or fewer parts. We give a conjecture characterizing which vectors are extreme in the cone of log concavity. We prove this characterization in one direction and give partial results in the other direction.