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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?

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    $\begingroup$ This may be a bit of a dumb question: what is the topology you are putting on $X_*$? And should the first appearance of the symbol $X_2$ be in fact $X_1$? $\endgroup$ – Willie Wong May 15 '14 at 10:09
  • $\begingroup$ Also, the paper you linked to specified finite subsets, 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
  • $\begingroup$ Lastly, what do you mean by "1-1 correspondence"? What about the subset of 2 points in $X_1$? On that component there is no extrema of the Morse function, because you exclude existence of antipodal points. Similar for all even numbers. $\endgroup$ – Willie Wong May 15 '14 at 10:23
  • $\begingroup$ @Willie, thanks, that was a typo and should have been $X_1$ in the first paragraph. These spaces have a natural metric defined by the Hausdorff distance between subsets, so the underlying topology is automatic. $\endgroup$ – Mikhail Katz May 15 '14 at 17:22
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    $\begingroup$ @katz: it would be better if you edited your question to incorporate these clarifications, rather than leaving them as comments. $\endgroup$ – Neil Strickland May 15 '14 at 21:01

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