Abstract
Ifv r is ther-dimensional volume of ther-simplex formed byr+1 points taken at random from a compact setK in ℝn, withr≤n, 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<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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Pfiefer, R.E. Maximum and minimum sets for some geometric mean values. J Theor Probab 3, 169–179 (1990). https://doi.org/10.1007/BF01045156
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DOI: https://doi.org/10.1007/BF01045156