F-Convexity of Real Functions

Let

be a family of sets in

\mathbb{R}^d R d

. A set

M \subset \mathbb{R}^d M ⊂ R d

is called

\cal F F

-convex if for any points

x,y\in M x , y ∈ M

there is a set

F\in \cal F F ∈ F

such that

x,y\in F x , y ∈ F

and

F\subset M F ⊂ M

. A function

f: \mathbb{R}\rightarrow \mathbb{R} f : R → R

is called

\cal F F

-convex if its epigraph is

\cal F F

-convex. In this paper we investigate various

\cal F F

-convexities for real functions