Extreme Convex Set Functions With Many Nonnegative Differences

. Where N is a finite set of the cardinality n and P the family of all its subsets, we study real functions on P having nonnegative differences of orders n \Gamma 2, n \Gamma 1 and n. Nonnegative differences of zeroth order, first order, and second order may be interpreted as nonnegativity, nonincreasingness and convexity, respectively. If all differences up to order n of a function are nonnegative, the set function is called completely monotone in analogy to the continuous case. We present a discrete Bernstein-type theorem for these functions with Mobius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative differences up to the orders n, n \Gamma 1 and n \Gamma 2, which is the most sophisticated case, and their Mobius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements. 1. Introduction Let N be a finite basic set of the cardinality n = jN j and P s r (N ..