Set-valued mapping (multifunction), domain, graph, and convexity

Let $X,Y$ be vector spaces. A set-valued mapping (or multifunction) is a map

that assigns to each $x\in X$ a (possibly empty) subset $F(x)\subset Y$.

• The domain of $F$ is $\mathrm{dom}(F):=\{x\in X: F(x)\neq\emptyset\}.$
• The graph of $F$ is the subset of the product space $X\times Y$ given by $\mathrm{gph}(F):=\{(x,y)\in X\times Y: y\in F(x)\}.$

The multifunction $F$ is called convex if its graph $\mathrm{gph}(F)$ is a convex set in $X\times Y$.

Context. Convex multifunctions generalize convex sets (as graphs) and appear in variational analysis and optimization (e.g., feasible-set and solution mappings).

Examples:

• Single-valued affine maps are convex multifunctions: if $F(x)=\{Ax+b\}$, then $\mathrm{gph}(F)$ is an affine subset of $X\times Y$.
• In $X=Y=\mathbb{R}$, the map $F(x)=[x,\infty)$ has a convex graph.
• The constant map $F(x)=C$ has $\mathrm{gph}(F)=X\times C$, convex iff $C$ is convex.