17 Sep
2013
17 Sep
'13
3:43 p.m.
Better upper bound: divide D-space into D+1 equal regions by "coning off" the D+1 faces of a regular D-simplex centered at 0, to the origin. If every region is occupied, then there can be no empty hemiball. (Because any hyperplane passing thru ball-center must contain at least one entire region on one side.)
The chance that all D+1 regions are occupied is lower bounded by D^N / (D+1)^N <= 1-Pempty.
--wrong. I should have said cone off the 2*D faces of an 0-centered hypercube to get 2*D regions. Need a centrosymmetric thing like hypercube; the regular D-simplex not being centrosymmetric, does not work. Then the corrected chance that all 2*D regions are occupied is lower bounded by (2*D-1)^N / (2*D)^N <= 1-Pempty.