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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Upper bounds for the volume and diameter of $ m$-dimensional sections of convex bodies


Author: Jesús Bastero
Journal: Proc. Amer. Math. Soc. 135 (2007), 1851-1859
MSC (2000): Primary 46B20, 52A20, 52A40
Published electronically: January 5, 2007
MathSciNet review: 2286096
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Abstract: In this paper some upper bounds for the volume and diameter of central sections of symmetric convex bodies are obtained in terms of the isotropy constant of the polar body. The main consequence is that every symmetric convex body $ K$ in $ \mathbb{R}^n$ of volume one has a proportional section $ K\cap F$, dim$ F= \lambda n$ ( $ 0< \lambda < 1$), of diameter bounded by

$\displaystyle R(K\cap F)\leq \frac {Cn^{3/4}\log (n+1)}{(1-\lambda)^3L_{K^\circ}}\, , $

whenever the polar body $ K^\circ$ is in isotropic position ($ C>0$ is some absolute constant).


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08693-5
PII: S 0002-9939(07)08693-5
Keywords: Asymptotic geometric analysis, diameter of sections
Received by editor(s): September 14, 2005
Received by editor(s) in revised form: February 8, 2006, and February 14, 2006
Published electronically: January 5, 2007
Additional Notes: The author was partially supported by DGA (Spain), MCYT (Spain) MTM2004-03036 and EU Project MRTN-CT-2004-511953 and is grateful to the Erwin Schrödinger Institut in Wien, where part of this work was carried out
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.