Complete Log Concavity of Coverage-Like Functions

We introduce an expressive subclass of non-negative almost submodular set functions, called strongly 2-coverage functions which include coverage and (sums of) matroid rank functions, and prove that the homogenization of the generating polynomial of any such function is completely log-concave, taking a step towards characterizing the coefficients of (homogeneous) completely log-concave polynomials. As a consequence we obtain that the "level sets" of any such function form an ultra-log concave sequence.