# 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

, Volume 4 (2) – Oct 25, 2004
18 pages

/lp/springer_journal/distances-between-non-symmetric-convex-bodies-and-the-mm-estimate-RlyDJGYLaf
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009842406728
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

PositivitySpringer Journals

Published: Oct 25, 2004

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