Relation between a convex set and convex function?

Question

I was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is called a convex function or something on those lines. (Note: I don't know whether the above line is true or if it makes any sense. I wrote it to give an idea of what I'm expecting.) I also found a few questions and answers in this site, but they are for some specific examples. I am looking for a general relation.

Any explanation is highly appreciated. Thanks!

Answer 1

If $f$ is convex, then the set $\{(x,y):y>f(x), a\leq x\leq b\}$ is convex.

Answer 2

In general, let $f \colon \mathbb{R}^{N} \to \left[ - \infty, + \infty \right]$. Then $f$ is convex in the sense that $$(\forall x,y \in \operatorname{dom}f)(\forall \alpha \in \left]0,1\right[)\quad f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y)$$ (here $\operatorname{dom}f = \left\{ x \in \mathbb{R}^{N} : f(x) < +\infty\right\}$) if and only if its epigraph $$\operatorname{epi}f = \left\{ (x,\xi ) \in \mathbb{R}^{N} \times \mathbb{R} : f(x) \leq \xi \right\}$$ is convex. There are many references for this, e.g., the book Convex Analysis and Monotone Operator Theory in Hilbert Spaces by Bauschke and Combettes, Chapter 8 (Definition 8.1 and Proposition 8.4). This is a nice connection between convex functions and convex sets. In particular, you can prove some properties of convex functions by working with their epigraphs, which are convex sets and sometimes easier to deal with.