Abstract
Ifvr is ther-dimensional volume of ther-simplex formed byr+1 points taken at random from a compact setK in ℝ n , withr≤n, andh is a (strictly) increasing function, then the (unique) compact set that gives the minimum expected value ofh o vr, is proved to be the ellipsoid (whenr=n) and the ball (whenr<n) almost everywhere. This result is established by using a single integral inequality for centrally symmetric quasiconvex functions integrated over compact rectangles.
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