Distance function to a set

d_Ω(x)=inf{||x−w||: w∈Ω} in a normed space

Distance function to a set

Let $(X,\|\cdot\|)$ be a normed vector space and let $\Omega\subset X$ be nonempty. The distance function to $\Omega$ is the real-valued map $d_\Omega:X\to[0,\infty)$ defined by

d_\Omega(x):=\inf\{\|x-w\|: w\in \Omega\}.

Context. Distance functions quantify constraint violation and are fundamental in projection methods and variational analysis.

Basic properties.

$d_\Omega(x)=0$ iff $x$ belongs to $\overline{\Omega}$ .
$d_\Omega$ is 1-Lipschitz: $|d_\Omega(x)-d_\Omega(y)|\le \|x-y\|$ .

Convexity. If $\Omega$ is convex , then $d_\Omega$ is convex. Conversely, if $\Omega$ is closed and $d_\Omega$ is convex, then $\Omega$ is convex (standard exercise-level fact).

Examples:

If $\Omega=\{0\}$ , then $d_\Omega(x)=\|x\|$ .
If $\Omega$ is a closed ball, $d_\Omega(x)$ is the excess of $\|x-x_0\|$ over the radius, truncated below by 0.