On a Polygon Equality Problem

Bernius and Blanchard of Bielefeld University in Germany have conjectured the followingpolygon inequality: for any two sets of vectorsx₁,...,x_nandy₁,...,y_nin R^m,[formula]in the 2-norm and that, moreover, equality holds in (*) if and only if there exists a permutation π on {1,2,...,n} such thaty_i=x_π(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 spaceX=L₁(Ω,μ), and subsequently use this characterization to complete the proof of the Bernius-Blanchard conjecture concerning the equality case in a Hilbert space.