Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the *diameter* of the pointset. Namely, this is the maximal distance among pairs of points of the set. Then the extrema of the Morse function are precisely the sets of vertices of odd regular polygons, and they are in 1-1 correspondence with the connected components of $X_1$ (each odd polygon is the unique critical point in its connected component).

A similar description holds for the space $X_2$ of subsets of the unit 2-sphere, of diameter smaller than that of an equilateral triangle inscribed in the equator. Again there are infinitely many connected components, and each component contains a unique extremum. This was proved here. Here an extremum is of a very special "selfdual" form, e.g., its convex hull has as many vertices as faces.

It is obvious how to generalize this to $n$ dimensions but I will stick with the 3-sphere. Thus, consider the space $X_3$ of subsets of the unit 3-sphere of diameter smaller than that of the regular tetrahedron inscribed in an equatorial 2-sphere. Here again there are infinitely many connected components. Is it true that there is a unique extremum in each?

finitesubsets, not arbitrary closed subsets as in your question statement, can you please clarify? (I ask because, relative to the Hausdorff metric if you allow arbitrary closed subsets I don't see how you get different "connected components"). $\endgroup$ – Willie Wong May 15 '14 at 10:16