Let A1β,...Anβ be operators acting on a separable complex Hilbert space
such that βi=1nβAiβ=0. It is shown that if A1β,...Anβ belong to a
Schatten p-class, for some p>0, then 2^{p/2}n^{p-1} \sum_{i=1}^n
\|A_i\|^p_p \leq \sum_{i,j=1}^n\|A_i\pm A_j\|^p_p for 0<pβ€2, and the
reverse inequality holds for 2β€p<β. Moreover, \sum_{i,j=1}^n\|A_i\pm
A_j\|^2_p \leq 2n^{2/p} \sum_{i=1}^n \|A_i\|^2_p for 0<pβ€2, and the
reverse inequality holds for 2β€p<β. These inequalities are related
to a characterization of inner product spaces due to E.R. Lorch.Comment: Minor revision, to appear in Math. Inequal. Appl. (MIA