I'm still a little bit stuck on the whole "two subsets of equal measure" line. "Any" hyperplane could leave one or more degenerate subsets untouched, leaving A_i whole, even though it does have measure zero.
In particular, I'm thinking of D=3, where we have unit circles in the xy-plane, centered at, say, (2,0,0), (0,2,0), and (0,-2,0). The only hyperplane that touches all three subsets of R^3 is the xy plane. Other hyperplanes cannot cut at least one subset in half.
Then, being left with only the xy-plane as an option (for two subsets), does it divide each A_i? Can it divide each A_i? |
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