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

Convex bodies with few faces


Authors: Keith Ball and Alain Pajor
Journal: Proc. Amer. Math. Soc. 110 (1990), 225-231
MSC: Primary 52A40; Secondary 11H46, 52A20
MathSciNet review: 1019270
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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that it $ {u_1}, \ldots ,{u_n}$ are vectors in $ {{\mathbf{R}}^k},k \leq n,1 \leq p < \infty $ and

$\displaystyle r = {\left( {\frac{1}{k}{{\sum\limits_1^n {\left\vert {{u_i}} \right\vert} }^p}} \right)^{1/p}}$

then the volume of the symmetric convex body whose boundary functionals are $ \pm {u_1}, \ldots , \pm {u_n}$, is bounded from below as

$\displaystyle {\left\vert {\left\{ {x \in {{\mathbf{R}}^k}:\left\vert {\left\la... ... \leq 1{\text{ for every }}i} \right\}} \right\vert^{1/k}} \geq 1/\sqrt \rho r.$

An application to number theory is stated.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1019270-8
PII: S 0002-9939(1990)1019270-8
Article copyright: © Copyright 1990 American Mathematical Society