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