Extreme convex set functions with many nonnegative differences Original Research Article

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 - 2, n - 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 Möbius inversion in the place of Laplace one. Numbers of all extreme functions with nonnegative differences up to the orders n, n - 1 and n - 2, which is the most sophisticated case, and their Möbius transforms are found. As an example, we write out all extreme nonnegative nondecreasing and semimodular functions to the set N with four elements.