Positive margin around a compact subset of an open set

Statement

If $K\subset U\subseteq\mathbb R^n$ , with $K$ nonempty and compact and $U$ open, then some $\delta>0$ satisfies

\{x:\operatorname{dist}(x,K)<\delta\}\subset U.

If $U\ne\mathbb R^n$ , equivalently

\inf_{x\in K}\operatorname{dist}(x,\mathbb R^n\setminus U)>0.

This is a positive support margin.

Proof

The distance to the closed complement is continuous and strictly positive on $K$ . It attains a positive minimum by compactness. Any smaller positive number is a suitable $\delta$ . If $U=\mathbb R^n$ , every $\delta>0$ works. Empty $K$ causes no constraint.

Use in localization

Margins leave room for a cutoff to transition from one to zero before reaching a boundary. They also allow small translations or mollifications of compactly supported functions while keeping their supports in $U$ .