An integral-valued set function f:2 V Z is called polymatroid if it is submodular, non-decreasing, and f( )=0. Given a polymatroid function f and an integer threshold t>=1, let α=α(f,t) denote the number of maximal sets X V satisfying f(X)<t, let β=β(f,t) be the number of minimal sets X V for which f(X)>=t, and let n=|V|. We show that if β>=2 then α=<β ( l o g t ) / c , where c=c(n,β) is the unique positive root of the equation 1=2 c (n c / l o g β -1). In particular, our bound implies that α=<(nβ) l o g t for all β>=1. We also give examples of polymatroid functions with arbitrarily large t,n,α and β for which α>=β ( 0 . 5 5 1 l o g t ) / c . More generally, given a polymatroid function f:2 V Z and an integral threshold t>=1, consider an arbitrary hypergraph H such that |H|>=2 and f(H)>=t for all H H. Let S be the family of all maximal independent sets X of H for which f(X)<t. Then |S|=<|H| ( l o g t ) / c ( n , | H | ) . As an application, we show that given a system of polymatroid inequalities f 1 (X)>=t 1 ,...,f m (X)>=t m with quasi-polynomially bounded right-hand sides t 1 ,...,t m , all minimal feasible solutions to this system can be generated in incremental quasi-polynomial time. In contrast to this result, the generation of all maximal infeasible sets is an NP-hard problem for many polymatroid inequalities of small range.