Expected convex hulls, order statistics, and Banach space probabilities (Q1100795)

The following result is shown: if \(X_ 1,X_ 2,..\). are i.i.d. points in \(R^ d\) with finite expected norm then their common distribution is determined by the sequence \(K_ 1\subseteq K_ 2\subseteq..\). where \(K_ n\) is the expectation of the convex hull of \(X_ 1,...,X_ n.\) This generalizes a scalar result that goes back to \textit{W. Hoeffding} [Ann. Math. Stat. 24, 93-100 (1953; Zbl 0050.136)]. Remarks include an application to Gaussian measures on Banach spaces.