Discrete Convexity : Convexity for Functions Defined on Discrete Spaces

The concept of discrete convexity for a real-valued function defined on a discrete space is an extension of the convexity definition of continuous functions. The equivalence of discrete convexity to the conventional definition of increasing (non-decreasing) first forward differences of functions of single variables is established. A further extension of the discrete convexity with submodularity yields the concept of strong discrete convexity. A function with the property of strong discrete convexity has a positive semi-definite matrix of second forward differences