A note on the volume of sections of \(B_{p}^{n}\)
Denote \(B_p^n = \{ (x_1,\dots,x_n) \in \mathbb{R}^n \mid \sum_i | x_i| ^p \leq 1 \}\) the unit ball of the \(l_p^n\) norm. Let \(H \subset \mathbb{R}^n\) be any \((n-1)\)-dimensional hyperplane that passes through the origin. It was proved by \textit{M. Meyer} and \textit{A. Pajor} [J. Funct. Anal. 80, No. 1, 109--123 (1988; Zbl 0667.46004)] for the case \(p \geq 2\) or \(p=1\), and by \textit{M. Schmuckenschläger} [Proc. Am. Math. Soc. 126, No. 5, 1527--1530 (1998; Zbl 0896.52009)], and by \textit{J. Bastero, F. Galve, \textit A. Peña} and \textit{M. Romance} [Proc. Am. Math. Soc. 130, No. 1, 183--192 (2002; Zbl 0988.52009)] for the case \(1 < p < 2\), that \( \text{Vol}_{n-1}(B_p^n \cap H)^n \geq \text{Vol}_n(B_p^n)^{n-1}. \) The paper under discussion presents a unified proof of this inequality, valid for all \(p \geq 1\). The proof is related to the arguments in [Zbl 0896.52009, Zbl 0988.52009], and it involves an estimate of the isotropic constant of \(B_p^n\) through analysis of the gamma function and its logarithmic derivatives.
scientific article; zbMATH DE number 5083591
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