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Maximal sections and centrally symmetric bodies. (English) Zbl 1012.52008

Summary: Let \(d\geq 2\) and let \(K\subset \mathbb{R}^d\) be a convex body containing the origin \(0\) in its interior. For each direction \(\omega\), let the \((d-1)\)-volume of the intersection of \(K\) and an arbitrary hyperplane with normal \(\omega\) attain its maximum when the hyperplane contains \(0\). Then \(K\) is symmetric about \(0\).
The proof uses a linear integro-differential operator on \(S^{d-1}\), whose null-space needs to be, and will be determined.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
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