Upper bounds for the volume and diameter of $m$-dimensional sections of convex bodies
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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 $K\cap F$, $\text {dim} F= \lambda n$ ($0< \lambda < 1$), of diameter bounded by \[ R(K\cap F)\leq \frac {Cn^{3/4}\log (n+1)}{(1-\lambda )^3L_{K^\circ }} , \] whenever the polar body $K^\circ$ is in isotropic position ($C>0$ is some absolute constant).References
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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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1851-1859
- MSC (2000): Primary 46B20, 52A20, 52A40
- DOI: https://doi.org/10.1090/S0002-9939-07-08693-5
- MathSciNet review: 2286096