Convexity on a convex subset via extension

Let $X$ be a real vector space, let $\Omega\subseteq X$ be a nonempty convex set, and let $f:\Omega\to\mathbb R$ .

Define the extension $\widetilde f:X\to(-\infty,+\infty]$ by

\widetilde f(x)= \begin{cases} f(x), & x\in\Omega,\\ +\infty, & x\notin\Omega. \end{cases}

The function $f$ is convex on $\Omega$ if $\widetilde f$ is a convex function on $X$ . Equivalently, for all $x,y\in\Omega$ and $\lambda\in[0,1]$ ,

f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y).

Interpretation

Extending by $+\infty$ packages the domain constraint into an extended-real function without changing the convexity inequality on $\Omega$ .