An isoperimetric inequality for hyperplane sections of a convex body
Let \(K\subset {\mathbb R}^n\) be a compact convex set, and let \(H\) be a hyperplane, orthogonal to the unit vector \(u\), which cuts \(K\) into two pieces, \(K_1\) and \(K_2\). Further, let \({\mathcal H}^i\) denote the \(i\)-dimensional Hausdorff measure. Then the main result of the paper is the inequality \[ {\mathcal H}^n(K_1){\mathcal H}^n(K_2)\leq (\ln 2)^{-1}\int_K|u\cdot(x-y)|dx {\mathcal H}^{n-1}(K\cap H) , \] which holds for all \(y\in{\mathbb R}^n\); moreover, if \(K\) is centrally symmetric, then it is shown that the factor \(1/\ln 2\) can be dropped and clearly the best estimate is obtained if \(y\) is the centre of symmetry of \(K\). The inequality for general \(K\) improves a result of \textit{R. Kannan, L. Lovász} and \textit{M. Simonovits} [Discrete Comput. Geom. 13, No. 3-4, 541-559 (1995; Zbl 0824.52012)] in the particular case where \(K\) is divided by a hyperplane. The idea of proof is to rewrite the inequalities in terms of the volumes of the hyperplane sections of \(K\) orthogonal to \(u\) and to use a crucial lemma, due to Kannan, Lovász and Simonovits, for logarithmically concave functions.
scientific article; zbMATH DE number 1335052
scientific article