Necessary conditions for a function to be convex: $\textbf{dom}f$ AND $\textbf{epi}f$ must be convex sets? Do they imply each other?

Question

According the p. 67 of Boyd's book on convexity, a function $f: \mathbb{R}^n \to \mathbb{R}$ is convex if $\textbf{dom}f$ is a convex set and if for all $x, y \in \textbf{dom}f$, and $\theta$ with $0 \le \theta \le 1$, we have $$ f(\theta x+ (1-\theta)y)\le \theta (f(x) + (1-\theta)f(y) $$ Moreover according to wikipedia, a function $f$ is convex if its epigraph is convex. Does those conditions automatically imply each other and, if yes, how can we prove it?

Answer 1

Yes, these are equivalent definitions. Note that $f(\theta x + (1-\theta) y) \le \theta f(x) + (1-\theta) f(y)$ where $0 \le t \le 1$ says that $\theta (x, f(x)) + (1-\theta) (y, f(y))$ is in the epigraph of $f$, and then of course so is $\theta (x,s) + (1-\theta) (y,t)$ if $s \ge f(x)$ and $t \ge f(y)$.