Extremalität von Ellipsoiden und die Faltungsungleichung von Sobolev (Q1057498)

In connection with the Riesz-Sobolev convolution inequality, \(f*g*h(0)\leq f^**g^**h^*(0)\) \((f^*\) is the Schwarz-symmetrization of f), the following characterization of ellipsoids is proved: Among all convex bodies \(A\subset {\mathbb{R}}^ n\) with given measure \(| A| >0\) the integral \(\int | A\cap (x-A)|^ pdx (1<p<\infty)\) is maximal iff A is an ellipsoid. In solving the problem of uniqueness the distribution function of \(A*A(x)=| A\cap (x-A)|\) is considered and its relation to the volume of the polar reciprocal of the projection body of A is exhibited.