Definition:Point-to-Set Distance

Definition
Let $\XX$ be a normed space.

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


 * $d: \XX \times \powerset \XX \to \hointr 0 \to$

where $\powerset \XX$ is the power set of $\XX$.

This mapping is defined as:


 * $\map d {x, C} := \inf \set {\norm {x - y}: y \in C}$

with the convention that for any $x \in \XX$:


 * $\map d {x, \O} = +\infty$

or, to be more mathematically rigorous:


 * $\forall C \in \powerset \XX \setminus \set \O: \map d {x, \O} > \map d {x, C}$