In a course on continuous optimization, the professor said that Chebyshev's problem is the following:
Let there be the unique projection onto a set $C$ for every point $x$ in a Hilbert space $X$ over the real numbers. Then, the set $C$ is convex.
Here the projection means a vector $p$ in the space such that
$$ \| x - p \| = \inf_{x \in X} \| x - C \| $$
Where can I find some results? My professor told me it is true in the Euclidean space, but I cannot see why.
