Estimates for the extremal sections of \(\ell ^n_p\)-balls
Let \(p\in [1, +\infty]\), \(B_p^n\) denote the unit ball of \(\ell_p^n\), \(r_{n,p}= (\text{vol}_n(B_p^n))^{-1/n}\), so \(r_{n,p}B_p^n\) is the normalized volume \(\ell ^n_p\)-ball. The main result of the paper is the upper estimate by \(\sqrt{\pi e}\) of the \((n-1)\)-dimensional volume of every central hyperplane section of the normalized \(\ell ^n_p\)-ball. In the proof the authors show by direct calculation that the isotropic constant of \(B_p^n\) takes its smallest value when \(p=2\) and then apply the well-known estimates of the hyperplane sections through the corresponding isotropic constant.
scientific article; zbMATH DE number 5848915
scientific article