Is the following set convex?

Question

The set of points closer to a given point than a given set, i.e.,
{x | ||x − x0|| ≤ ||x − y|| for all y ∈ S}
where S ⊆ R^n

It reminds me of an Euclidean ball, but I don't know how to proceed to check if it's convex or not.
( In this case it's the 2-norm used above ).

Answer 1

The link: https://see.stanford.edu/materials/lsocoee364a/hw1sol.pdf
2.7 provides the answer, where you prove that the set is a halfspace and thus the set is convex.