Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

Convex analysis is the study of properties of convex sets and convex functions.

Formally, a set $S$ is *convex* if, for all $x, y \in S$ and $t \in [0,1]$, $tx + (1-t)y \in S$. Intuitively, this says that for any two points in $S$, the line segment connecting those points is also in $S$.

Formally, a function $f: S \to \mathbb{R}$ defined on a convex set $S$ is *convex*, if for all $x,y \in S$ and $t \in [0,1]$, $f(tx + (1-t)y) \leq t f(x) + (1-t) f(y)$. Intuitively, this says that, for any two points on the graph of $f$, the line segment connecting those points lies above the graph of $f$.

The classic reference in this subject is R.T. Rockafellar, *Convex Analysis*.

Some of the more important results in convex analysis include Carathéodory's Theorem, Jensen's Inequality, Minkowski's Theorem, and the supporting hyperplane theorem.