## Voronoi tesselation for a pair of convex sets

Let $(M, d)$ be a Euclidean metric space and $X$ and $Y$ be distinct non-empty subsets of $M$. Let $S_{X}$ be the set of elements $x$ of $M$ for which the Hausdorff distance between $x$ and $X$ is smaller than the Hausdorff distance between $x$ and $Y$, i.e., $\lbrace x \vert d(x,X) \textless{} d(x,Y) \rbrace$. Is $S_{X}$ convex? Or, if there is no general answer to this, is the number of dimensions here essential?