Discrete & Computational Geometry

, Volume 55, Issue 1, pp 185–202 | Cite as

On the Complexity of the Set of Unconditional Convex Bodies

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Abstract

We show that for any \(1 \le t \le \tilde{c} n^{1/2} \log ^{-5/2} n\), the set of unconditional convex bodies in \(\mathbb {R}^n\) contains a t-separated subset of cardinality at least
$$\begin{aligned} \exp \Big (\exp \Big (\frac{c}{t^2 \log ^4 (1+t)} \, n \Big ) \Big ). \end{aligned}$$
This implies the existence of an unconditional convex body in \(\mathbb {R}^n\) which cannot be approximated within the distance d by a projection of a polytope with N faces unless \(N \ge \exp (c(d) n)\). We also show that for \(t \ge 2\), the cardinality of a t-separated set of completely symmetric bodies in \(\mathbb {R}^n\) does not exceed
$$\begin{aligned} \exp \Big (\exp \Big (C \frac{\log ^2 n}{\log t} \Big ) \Big ). \end{aligned}$$
.

Keywords

Unconditional convex bodies Approximation by polytopes  Banach–Mazur distance 
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Notes

Acknowledgments

The author is grateful to Olivier Guédon for several suggestions which allowed to clarify the presentation. This study was partially supported by NSF Grants DMS-01161372, DMS-1464514, and USAF Grant FA9550-14-1-0009.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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