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Extremalität von Ellipsoiden und die Faltungsungleichung von Sobolev. (English) Zbl 0563.52009
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.

MSC:
52A22 Random convex sets and integral geometry (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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