Definition:Point-to-Set Distance

Definition
Let $\mathcal X$ be a normed space.

The point-to-set distance on $\mathcal X$ is a mapping:


 * $d: \mathcal X \times \mathcal P \left({\mathcal X}\right) \to \left[{0 \,.\,.\, \infty}\right)$

where $P \left({\mathcal X}\right)$ is the power set of $\mathcal X$.

This mapping is defined as:


 * $d \left({x, C}\right) := \inf \left\{{\left\|{x - y}\right\|: y \in C}\right\}$

with the convention that for any $x \in \mathcal X$:


 * $d \left({x, \varnothing}\right) = +\infty$

or, to be more mathematically rigorous:


 * $\forall C \in \mathcal P \left({\mathcal X}\right) \setminus \left\{{\varnothing}\right\}: d \left({x, \varnothing}\right) > d \left({x, C}\right)$