Convexity on a convex subset via extension

Define convexity on Ω by extending f to X with value ∞ outside Ω

Convexity on a convex subset via extension

Let $X$ be a vector space, let $\Omega\subset X$ be a nonempty convex set , and let $f:\Omega\to\mathbb{R}$ be a real-valued function.

Define the extension $\tilde f:X\to\mathbb{R}$ by

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

We say that $f$ is convex on $\Omega$ if $\tilde f$ 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).

Context. This convention packages “convexity + domain restriction” into a single extended-real function, aligning convexity on subsets with epigraph-based convexity.

Example. If $\Omega$ is the unit ball in a normed space and $f(x)=\|x\|$ on $\Omega$ , then $\tilde f$ encodes the constraint $x\in\Omega$ by assigning $\infty$ outside.