  Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space
such that $\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ 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 $0<p\leq 2$, and the
reverse inequality holds for $2\leq p<\infty$. 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 $0<p\leq 2$, and the
reverse inequality holds for $2\leq p<\infty$. These inequalities are related
to a characterization of inner product spaces due to E.R. Lorch.
