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Inequalities for the Gamma function and estimates for the volume of sections of $B^n_p$


Authors: Jesús Bastero, Fernando Galve, Ana Peña and Miguel Romance
Journal: Proc. Amer. Math. Soc. 130 (2002), 183-192
MSC (2000): Primary 52A21, 33B15; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-01-06139-1
Published electronically: June 8, 2001
MathSciNet review: 1855637
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Abstract:

Let $B^n_p=\{(x_i)\in\mathbb{R}^n;\sum_1^n\vert x_i\vert^p\leq1\}$ and let $E$ be a $k$-dimensional subspace of $\mathbb{R}^n$. We prove that $\vert E\cap B^n_p\vert _k^{1/k}\geq \vert B^n_p\vert _n^{1/n}$, for $1\leq k\leq (n-1)/2$and $k=n-1$ whenever $1<p<2$. We also consider $0<p<1$ and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.


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Additional Information

Jesús Bastero
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: bastero@posta.unizar.es

Fernando Galve
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: two@maths.univ.edu.au

Ana Peña
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: anap@posta.unizar.es

Miguel Romance
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: mromance@posta.unizar.es

DOI: https://doi.org/10.1090/S0002-9939-01-06139-1
Keywords: Gamma function, inequalities, sections of convex bodies
Received by editor(s): May 31, 2000
Published electronically: June 8, 2001
Additional Notes: The first, the third and the fourth authors were supported in part by a DGES Grant (Spain).
The fourth author was also supported by an FPI Grant (Spain).
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2001 American Mathematical Society

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