Distances Between Non-symmetric Convex Bodies and the $$MM^* $$ -estimate

Distances Between Non-symmetric Convex Bodies and the $$MM^* $$ -estimate Let K, D be n-dimensional convex bodes. Define the distance between K and D as $$d(K,D) = \inf \{ \lambda |TK \subset D + x \subset \lambda \cdot TK\} ,$$ where the infimum is taken over all $$x \in {\mathbb{R}}^n $$ and all invertible linear operators T. Assume that 0 is an interior point of K and define $$M(K) = \smallint _{S^{n - 1} } |\omega |_K d\mu (\omega ),$$ where μ is the uniform measure on the sphere. We use the difference body estimate to prove that K can be embedded into $${\mathbb{R}}^n $$ so that $$M(K) \cdot M(K^ \circ ) \leqslant Cn^{1/3} \log ^a n$$ for some absolute constants C and $$a$$ . We apply this result to show that the distance between two n-dimensional convex bodies does not exceed $$n^{4/3} $$ up to a logarithmic factor. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Distances Between Non-symmetric Convex Bodies and the $$MM^* $$ -estimate

Positivity , Volume 4 (2) – Oct 25, 2004

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Kluwer Academic Publishers
Copyright © 2000 by Kluwer Academic Publishers
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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