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On a generalization of the Hermite-Hadamard inequality and applications in convex geometry. (English) Zbl 1447.52007

Summary: In this paper, we show the following result: if \(C\) is an \(n\)-dimensional 0-symmetric convex compact set, \(f:C\rightarrow [0,\infty)\) is concave, and \(\phi :[0,\infty)\rightarrow [0,\infty)\) is not identically zero, convex, with \(\phi (0)=0\), then \[ \frac{1}{|C|} \int_C\phi (f(x))\mathrm{d} x\leq \frac{1}{2}\int_{-1}^1\phi (f(0)(1+t))\mathrm{d}t, \] where \(|C|\) denotes the volume of \(C\). If \(\phi\) is strictly convex, equality holds if and only if \(f\) is affine, \(C\) is a generalized symmetric cylinder and \(f\) becomes 0 at one of the basis of \(C\). We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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