On a Polygon Equality Problem

Bernius and Blanchard of Bielefeld University in Germany have conjectured the following polygon inequality: for any two sets of vectors x1, . . . , xn and y1, . . . , yn in Rm, n ). 5xiyxj5q 5yiyyj5F 5xiyyj5 i-j i-j i, js1 in the 2-norm and that, moreover, equality holds in ). if and only if there exists a permutation p on 1, 2, . . . , n4such that yisxp i., is1, . . . , 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 XsL1 V, m., and subsequently use this characterization to complete the proof of the Bernius]Blanchard conjecture concerning the equality case in a Hilbert space.