F-Convexity of Real Functions

Let $\cal F$ be a family of sets in $\mathbb{R}^d$. A set $M \subset \mathbb{R}^d$ is called $\cal F$-convex if for any points $x,y\in M$ there is a set $F\in \cal F$ such that $x,y\in F$ and $F\subset M$. A function $f: \mathbb{R}\rightarrow \mathbb{R}$ is called $\cal F$-convex if its epigraph is $\cal F$-convex. In this paper we investigate various $\cal F$-convexities for real functions