Inequalities for the Gamma function and estimates for the volume of sections of \(B^n_p\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 0988.52009
DOI: 10.1090/S0002-9939-01-06139-1
author: Jesús Bastero
author: Fernando Galve
author: A. Peña
author: Miguel Romance
published in: Proceedings of the American Mathematical Society
publication date: 2001-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0008007
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1663148
zbMATH Keywords: sections of convex bodies
zbMATH Keywords: inequalities
zbMATH Keywords: Gamma function
MaRDI profile type: Publication
arXiv ID: math/0008007
cites work: Shadows of Convex Bodies
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cites work: An application of the Fourier transform to sections of star bodies
cites work: Sections of the unit ball of \(\ell ^ n_ p\)
cites work: Q4728828
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cites work: Q4370600
cites work: Volume of intersections and sections of the unit ball of $\ell ^n_p$
cites work: A geometric inequality with applications to linear forms
OpenAlex ID: W1847807595
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title: Inequalities for the Gamma function and estimates for the volume of sections of \(B^n_p\)
review text: 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.
reviewed by: Mikhail Ostrovskii
description: scientific article; zbMATH DE number 1663148