Extremal cross-polytopes and Gaussian vectors

Let C = C(l_1, ..., l_n) be the n-dimensional orthogonal cross-polytope whose axes are of length l_1,..., l_n. Subject to the condition \sum l_i^2 = 1, the mean width of C is minimised when l_i = 1/sqrt{n} for every i, and it is maximised when C is at most two dimensional. As a corollary, a lower bound on the mean width of a general convex body K is derived in terms of the successive inner radii of K. A more general result is presented for Gaussian random vectors.

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