Overview
The closest point within a set to a subset of that set, is the point that has the smallest distance to that subset. (formal definition given below)
Theorem
For a closed convex subset {% A %} of a Banach Space H, and for a point {% x \in H %} but not in {% A %}, there exists a unique {% a \in A %} such that
{% || x- \hat{a} || = inf{||x-a|| : a \in A} %}
Here, we identify
{% distance(x,A) = inf{||x-a|| : a \in A} %}