Let $N$ be a finite set and $2^N$ be the set of all subsets of $N$. Consider the function $f:2^N\rightarrow \mathbb{R}$.
$f$ is said to be convex if for any two $Q,S\in 2^N$ $$f(Q\cup S)\ge f(Q)+f(S)-f(Q\cap S).$$
However, I cannot recognize any connection with the notion of convexity of a function defined on the real line. Can you give any clue on how to interpret this particular sort of convexity?
UPDATE: As @Christoph rightfully noted, the described function is called supermodular (there is no entry on this kind of supermodular functions in Wikipedia, but I saw it elsewhere).
However, I'm still puzzled about the relation between sub(super)modular set functions and concave (convex) functions. In the article on submodular functions it is said that "Submodular functions have properties which are very similar to convex and concave functions..."
Does it mean that they are both convex and concave? What are these properties?