Inequalities for the Gamma function and estimates for the volume of sections of \(B^n_p\) (Q2750904)

Let \(B^n_p=\{(x_i) \in\mathbb{R}^n: \sum^n_1 |x_i|^p \leq 1\}\), \(0<p <\infty\), and let \(E\) be a \(k\)-dimensional subspace of \(\mathbb{R}^n\). By \(|A|_k\) the authors denote the \(k\)-dimensional volume of \(A\).NEWLINENEWLINENEWLINE\textit{J. D. Vaaler} [Pac. J. Math. 83, 543-553 (1979; Zbl 0465.52011)] proved a result that can be restated as the inequality NEWLINE\[NEWLINE|E\cap B_p^n |_k^{1/k} \geq|B_p^n |_n^{1/n} \tag{1}NEWLINE\]NEWLINE in the case \(p=\infty\).NEWLINENEWLINENEWLINE\textit{M. Meyer} and \textit{A. Pajor} [J. Funct. Anal. 80, No. 1, 109-123 (1988; Zbl 0667.46004)] extended Vaaler's result to the range \(p\in\{1\} \cup [2, \infty]\). NEWLINENEWLINENEWLINEFrom the text: ``More recently \textit{M. Schmuckenschläger} [Proc. Am. Math. Soc. 126, No. 5, 1527-1530 (1998; Zbl 0896.52009)] estimated the volume of the \((n-1)\)-dimensional sections of \(B_p^n\), for \(1<p< 2\), but the proof of the inequality he proposed was not correct.''NEWLINENEWLINENEWLINEThe aim of this paper is to give a proof of the inequality appearing in Schmuckenschläger's paper cited above, to prove (1) for \(1\leq k\leq (n-1)/2\) and \(1<p <2\), to prove an analogue of (1) for \(0<p<1\), and to prove (1) for \(B=B^n_2 \oplus_p B^n_2\), \(1\leq p\leq 2\) answering a question raised to the authors by M. Meyer.NEWLINENEWLINENEWLINEIn order to do this the authors establish sharp inequalities involving the Gamma function which are of interest by themselves.