, Volume 3, Issue 2, pp 169-179

Maximum and minimum sets for some geometric mean values

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Abstract

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