  In this paper we prove that the ball is a maximizer of the Schatten $p$-norm
of some convolution type integral operators with non-increasing kernels among
all domains of a given measure in $\mathbb R^{d}$. We also show that the
equilateral triangle has the largest Schatten $p$-norm among all triangles of a
given area. Some physical motivations for our results are also presented.
