On Contact Points of Convex Bodies

نویسنده

  • Nikhil Srivastava
چکیده

We show that for every convex body K in R, there is a convex body H such that H ⊂ K ⊂ c ·H with c = 2.24 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 [5], 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.

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تاریخ انتشار 2009