Geometric Aspects of Functional Analysis pp 393-412 | Cite as

# On Contact Points of Convex Bodies

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## Abstract

We show that for every convex body and

*K*in \({\mathbb{R}}^{n}\), there is a convex body*H*such that$$H \subset K \subset c \cdot H\qquad \qquad \text{ with}\ c = 2.24$$

*H*has at most*O*(*n*) contact points with the minimal volume ellipsoid that contains it. When*K*is symmetric, we can obtain the same conclusion for every constant*c*> 1. We build on work of Rudelson [Israel J. Math.**101**(1), 92–124 (1997)], who showed the existence of*H*with \(O(n\log n)\) contact points. The approximating body*H*is constructed using the “barrier” method of Batson, Spielman, and the author, which allows one to extract a small set of vectors with desirable spectral properties from any John’s decomposition of the identity. The main technical contribution of this paper is a way of controlling the*mean*of the vectors produced by that method, which is necessary in the application to John’s decompositions of nonsymmetric bodies.## Keywords

Convex Bodies Rudelson Non-symmetric Bodies Main Technical Contribution John Ellipsoid
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgements

The author wishes to thank Daniel Spielman for helpful comments as well as the Department of Computer Science at Yale University, where this work was done.

## References

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*An Elementary Introduction to Modern Convex Geometry*. Flavors of Geometry (University Press, 1997), pp. 1–58Google Scholar - 2.J.D. Batson, D.A. Spielman, N. Srivastava, in
*Twice-Ramanujan Sparsifiers*. STOC ’09: Proceedings of the 41st annual ACM symposium on Theory of computing (ACM, New York, 2009), pp. 255–262Google Scholar - 3.A. Giannopoulos, V. Milman, Extremal problems and isotropic positions of convex bodies. Israel J. Math.
**117**, 29–60 (2000)MathSciNetzbMATHCrossRefGoogle Scholar - 4.P.M. Gruber, Minimal ellipsoids and their duals. Rendiconti del Circolo Matematico di Palermo
**37**(1), 35–64 (1988)MathSciNetzbMATHCrossRefGoogle Scholar - 5.M. Rudelson, Contact points of convex bodies. Israel J. Math.
**101**(1), 92–124 (1997)MathSciNetCrossRefGoogle Scholar - 6.M. Rudelson, Random vectors in the isotropic position. J. Funct. Anal.
**163**(1), 60–72 (1999)MathSciNetCrossRefGoogle Scholar

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