For each positive integer n, let E(n) be n-dimensional Euclidean space-with its standard metric-and let M(n) be the separable metric space whose "points" are all the non-empty compact subsets of E(n) and whose "distance" is the Hausdorff distance between each pair of these subsets. Let F(n) be the mapping of M(n) onto E(n) which maps every non-empty compact subset of E(n) to its centroid. Is F(n) continuous at all "points" of M(n)?



A question about the centroids of compact subsets of Euclidean spaces
Even if you restrict to convex sets, I don't think there's a definition of centroid that makes this work for $n > 1$.


For instance, in $\mathbb{R}^2$ consider rectangles of the form $[0, 1] \times [0, \epsilon]$ versus triangles with vertices $(0, 0), (1, 0), (0, \epsilon)$.


As $\epsilon \to 0$, both families of shapes converge to the same line segment, but their centroids converge to different points: $(1/2, 0)$ and $(1/3, 0)$ respectively.


The best bet to rescue the theorem might be to restrict to the space of convex sets with positive measure.


