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Published in:
2012 | OriginalPaper | Chapter
On Contact Points of Convex Bodies
Author : Nikhil Srivastava
Published in: Geometric Aspects of Functional Analysis
Publisher: Springer Berlin Heidelberg
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Abstract
We show that for every convex body K in \({\mathbb{R}}^{n}\), there is a convex body H such that and 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.
$$H \subset K \subset c \cdot H\qquad \qquad \text{ with}\ c = 2.24$$