Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

A subset ${\displaystyle C\subseteq X}$ of some vector space ${\displaystyle X}$ is convex if it satisfies any of the following equivalent conditions:

Throughout, ${\displaystyle f:X\to [-\infty ,\infty ]}$ will be a map valued in the extended real numbers ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}$ with a domain ${\displaystyle \operatorname {domain} f=X}$ that is a convex subset of some vector space. The map ${\displaystyle f:X\to [-\infty ,\infty ]}$ is a convex function if

holds for any real ${\displaystyle 0<r<1}$ and any ${\displaystyle x,y\in X}$ with ${\displaystyle x\neq y.}$ If this remains true of ${\displaystyle f}$ when the defining inequality (Convexity ≤) is replaced by the strict inequality

then ${\displaystyle f}$ is called strictly convex.

Convex functions are related to convex sets. Specifically, the function ${\displaystyle f}$ is convex if and only if its epigraph

is a convex set. The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.

The domain of a function ${\displaystyle f:X\to [-\infty ,\infty ]}$ is denoted by ${\displaystyle \operatorname {domain} f}$ while its effective domain is the set