Vol. 45, No. 2, 1973

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On some mean values associated with a randomly selected simplex in a convex set

H. Groemer

Vol. 45 (1973), No. 2, 525–533

For any convex body K in euclidean n-space denote by m(K) the mean value of the volume of a simplex with vertices at n + 1 randomly selected points from K. It is shown that among all convex bodies of given volume the mean value m(K) is minimal if and only if K is an ellipsoid. Actually, a more general result is obtained which shows that the higher order moments of the volume of a randomly selected simplex in a convex set have similar minimal properties.

Mathematical Subject Classification 2000
Primary: 60D05
Secondary: 52A10
Received: 28 February 1972
Revised: 11 May 1972
Published: 1 April 1973
H. Groemer