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The only idea that comes to mind for improving @RobertIsrael's bound, following from Noam Elkies's answer, would be to integrate over all rectangles of area $c^2/\beta^2$, not just squares -- I'll see if I can make any improvements.
That is pretty awesome. So, the fact that this is an upper bound comes from the fact that the region whose volume you take doesn't have to lie in $\mathcal{S}^{n}$?
