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Positivity

, Volume 8, Issue 4, pp 401–406 | Cite as

On the Dual Form of ‘Low M* Estimate’ in the Quasi-convex Case

  • A. E. Litvak
Article
  • 35 Downloads

Abstract

Let \(B_{2}^{n}\) denote the Euclidean ball in \({\mathbb R}^n\), and, given closed star-shaped body \(K \subset {\mathbb R}^{n}, M_{K}\) denote the average of the gauge of K on the Euclidean sphere. Let \(p \in (0,1)\) and let \(K \subset {\mathbb R}^{n}\) be a p-convex body. In [17] we proved that for every \(\lambda \in (0,1)\) there exists an orthogonal projection P of rank \((1 - \lambda)n\) such that
$$\frac{f(\lambda)}{M_K} PB^{n}_{2} \subset PK,$$
where \(f(\lambda)=c_p\lambda^{1+1/p}\) for some positive constant c p depending on p only. In this note we prove that \(f(\lambda)\) can be taken equal to \(C_p\lambda^{1/p-1/2}\). In terms of Kolmogorov numbers it means that for every \(k \leq n\)
$$d_k (\hbox{Id}:\ell^{n}_{2} \to ({\mathbb R}^{n},\|\cdot\|_{K})) \leq C_p \frac{n^{1/p-1}}{k^{1/p-1/2}} \ell (\hbox{ID}: \ell^{n}_{2} \to ({\mathbb R}^{n}, \|\cdot\|_{K})),$$
where \(\ell(\hbox{Id})={\bf E}\|\sum\limits^{n}_{i=1}g_i e_i\|_K\) for the independent standard Gaussian random variables \(\{g_i\}\) and the canonical basis \(\{e_i\}\) of \({\mathbb R}^n\). All results do not require the symmetry of K.

Keywords

\(\ell\)-functional Kolmogorov numbers ‘low M*-estimate’ p-norms Quasi-convexity ‘random’ projections 

Mathematics Subject Classifications (1999). Primary:

46B07 52A30 46A16 

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Copyright information

© Kluwer Academic Publishers 2005

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, ABCanada

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