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Sections of convex bodies through their centroid. (English) Zbl 0901.52005

Let \(K\) be a convex body in \(\mathbb{R}^n,\) \(E^k\) a \(k\)-dimensional affine subspace of \(\mathbb{R}^n\) passing through the centroid of \(K.\) Denote by \(F(K,E^k)\) the ratio between the \(k\)-dimensional volume of the section of \(K\) by \(E^k\) and the \(k\)-dimensional volume of the maximal section of \(K\) parallel to \(E^k.\) Then \(F(K,E^k)\geq(\frac{k+1}{n+1})^k\) and the equality case is solved. For example, there is equality if \(K\) is a simplex with a \(k\)-face parallel to \(E^k.\) For \(k = 1,\) this estimate goes back to T. Bonnesen and W. Fenchel [Theory of convex bodies. BCS Associates, Moscow, ID (1987; Zbl 0628.52001); German original: Berlin (1934; Zbl 0008.07708)], or even further. For \(k = n - 1,\) the result is due to E. Makai and H. Martini [Geom. Dedicata 63, No. 3, 267-296 (1996; Zbl 0865.52005)].
Reviewer: S.M.Pokas (Odessa)

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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