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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.

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)
Full Text: DOI
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