PUB - Publikationen an der Universität Bielefeld

Bernius and Blanchard of Bielefeld University in Germany have conjectured the following polygon inequality: for any two sets of vectors x(1),..., x(n) and y(1),..., y(n) in R-m, [GRAPHICS] in the 2-norm and that, moreover, equality holds in(*)if and only if there exists a permutation pi on {1, 2,..., n} such that y(i) = x(pi(i)), i = 1,..., n. That (*) is valid is a consequence of an inequality that holds in certain Banach spaces and which was recently proved by Lennard, Tongue, and Weston. We therefore characterize here the case of equality in (*), actually for vectors in the space X = L-1(Omega, mu), and subsequently use this characterization to complete the proof of the Bernius-Blanchard conjecture concerning the equality case in a Hilbert space. (C) 1998 Academic Press.