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?