Let mu be an even compactly supported Borel probability measure on the real line. For every N > n consider N independent random vectors X-1,..., X (N) in R-n, with independent coordinates having distribution mu. We establish a sharp threshold for the volume of the random polytope K (N) := conv {X-1,..., X-N}, provided that the Legendre transform lambda of the cumulant generating function of mu satisfies the condition [GRAPHICS] where alpha is the right endpoint of the support of mu. The method and the result generalize work of Dyer, Furedi and McDiarmid on 0/1 polytopes. We verify (*) for a large class of distributions.
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