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A geometric inequality and a low \(M\)-estimate. (English) Zbl 1064.52003
Author’s abstract: We present an integral inequality connecting volumes and diameters of sections of a convex body. We apply this inequality to obtain some new inequalities concerning diameters of sections of convex bodies, among which is our “low \(M\)-estimate”. Also, we give novel, alternative proofs to some known results, such as the fact that a finite volume ratio body has proportional sections that are isomorphic to a Euclidean ball.

MSC:
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
46B20 Geometry and structure of normed linear spaces
52A40 Inequalities and extremum problems involving convexity in convex geometry
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