Convex Sets and Nearest Neighbors

A nonempty set $S$ in a normed linear space $X$ is called a Chebyshev set if for each $u \in X$ there is exactly one nearest point in $S$ to $u$ (i.e., for a Chebyshev set, nearest points always exist, and they are unique).

For finite dimensional spaces it is known that:

Theorem. A nonempty set in the Euclidean space $R^n$ is Chebyshev if and only if it is closed and convex.

In infinite dimensions the situation is much more subtle (and some aspects are still open).

Please have a look at this really nice overview talk which explains this, provides references to the state-of-the-art.

This is the celebrated Chebyshev problem. The answer is positive in $\mathbb{R}^n$, and still open in the Hilbert space.