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M-Estimates for Isotropic Convex Bodies and Their L q -Centroid Bodies

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

Let K be a centrally-symmetric convex body in \(\mathbb{R}^{n}\) and let \(\|\cdot \|\) be its induced norm on \(\mathbb{R}^{n}\). We show that if K ⊇ rB 2 n then:

$$\displaystyle{\sqrt{n}M(K)\leqslant C\sum _{k=1}^{n} \frac{1} {\sqrt{k}}\min \left (\frac{1} {r}, \frac{n} {k}\log \Big(e + \frac{n} {k}\Big) \frac{1} {v_{k}^{-}(K)}\right )}$$

where \(M(K) =\int _{S^{n-1}}\|x\|\,d\sigma (x)\) is the mean-norm, C > 0 is a universal constant, and v k −(K) denotes the minimal volume-radius of a k-dimensional orthogonal projection of K. We apply this result to the study of the mean-norm of an isotropic convex body K in \(\mathbb{R}^{n}\) and its L q -centroid bodies. In particular, we show that if K has isotropic constant L K then:

$$\displaystyle{M(K)\leqslant \frac{C\log ^{2/5}(e + n)} {\root{10}\of{n}L_{K}}.}$$

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Acknowledgements

The first named author acknowledges support from the programme “APIΣTEIA II” of the General Secretariat for Research and Technology of Greece. The second named author is supported by ISF (grant no. 900/10), BSF (grant no. 2010288), Marie-Curie Actions (grant no. PCIG10-GA-2011-304066) and the E. and J. Bishop Research Fund.

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Authors and Affiliations

  1. Department of Mathematics, University of Athens, Panepistimioupolis 157-84, Athens, Greece

    Apostolos Giannopoulos

  2. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel

    Emanuel Milman

Authors
  1. Apostolos Giannopoulos
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  2. Emanuel Milman
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Correspondence to Emanuel Milman .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Giannopoulos, A., Milman, E. (2014). M-Estimates for Isotropic Convex Bodies and Their L q -Centroid Bodies. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_13

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