convex set

Definition

A set $\mathcal{S}$ is convex if for any $\mathbf{x},\mathbf{y} \in \mathcal{S}$, $\lambda \in [0,1]$: $$(1-\lambda)\mathbf{x} + \lambda \mathbf{y} \in \mathcal{S}$$

Theorem:

Let $f : S \longrightarrow \mathbb{R}$ be a function defined on the convex subset $S$ of a real linear space $L$. Then, $f$ is convex on $S$ if and only if its epigraph is a convex subset of $S × \mathbb{R}$; $f$ is concave if and only if its hypograph is a convex subset of $S × \mathbb{R}$.

References

1. https://www.cs.umb.edu/~dsim/cs724/sconvs3.pdf