best approximation

One of the problems in approximation theory is to determine points that minimize distances (to a given point or subset). More precisely,

Problem - Let $X$ be a metric space and $S\subseteq X$ a subset. Given $x_0\in X$ we want to know if there exists a point in $S$ that minimizes the distance to $x_0$, i.e. if there exists $y_0\in S$ such that

$$d(x_0,y_0)=\underset{y\in S}{inf}d(x_0,y)$$

Definition - A point $y_0$ that the above conditions is called a best approximation of $x_0$ in $S$.

In general, best approximations do not exist. Thus, the problem is usually to identify classes of spaces $X$ and $S$ where the existence of best approximations can be assured.

Example : When $S$ is compact, best approximations of a given point $x_0\in X$ in $S$ always exist.

After one assures the existence of a best approximation, one can question about its uniqueness and how to calculate it explicitly.

Remark - There is no reason to restrict to metric spaces. The definition of best approximation can be given for pseudo-metric spaces, semimetric spaces or any other space where a suitable notion of ”distance” can be given.