Indicator function of a set

Let $X$ be a vector space and let $\Omega\subset X$ be nonempty. The indicator function of $\Omega$ is the extended-real-valued function $\delta_\Omega:X\to\mathbb{R}$ defined by

Its domain is $\mathrm{dom}(\delta_\Omega)=\Omega$, and its epigraph is $\mathrm{epi}(\delta_\Omega)=\Omega\times[0,\infty)$.

Context. Indicator functions encode constraints as penalties: minimizing $f+\delta_\Omega$ is equivalent to minimizing $f$ subject to $x\in\Omega$.

Convexity. $\delta_\Omega$ is convex if and only if $\Omega$ is a convex set .

Examples:

• If $\Omega$ is a subspace, then $\delta_\Omega$ is convex.
• If $\Omega$ is a nonconvex set (e.g., two disjoint balls), then $\delta_\Omega$ is not convex.