Recap on convex sets and functions

Convex set

$C\subseteq {\mathbb{R}}^n$ is a convex set if and only if:

$$\lambda \mathbf{x}+(1-\lambda )\mathbf{y}\in C,\forall \mathbf{x},\mathbf{y}\in C,\forall \lambda \in [0,1].$$

The condition $\lambda \in [0,1]$ means that we are taking all the points on the line segment joining $\mathbf{x}$ and $\mathbf{y}$, including the endpoints. Therefore, a set is convex exactly when the whole segment between any two of its points remains inside the set.

Epigraph

Let $f$ be a function from a set $S\subseteq {\mathbb{R}}^n$ to the extended real line

$$\overline{\mathbb{R}}=\mathbb{R}\cup \{+\infty \}\cup \{-\infty \}.$$

The epigraph of $f$ is the subset of ${\mathbb{R}}^{n+1}$ defined by

$$\mathrm{epi}(f)=\left\{(\mathbf{x},y)\in S\times \mathbb{R}:y\ge f(\mathbf{x})\right\}.$$

Geometrically, the epigraph is the set of all points lying on or above the graph of the function.

Convex function