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Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates

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

Abstract

Chaining techniques show that if X is an isotropic log-concave random vector in \(\mathbb{R}^{n}\) and Γ is a standard Gaussian vector then

$$\displaystyle{\mathbb{E}\Vert X\Vert \leq Cn^{1/4}\mathbb{E}\Vert \varGamma \Vert }$$

for any norm \(\Vert \cdot \Vert\), where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant

$$\displaystyle{\sigma _{n} =\sup {\Bigl ( \sqrt{\mathrm{Var }(\vert X\vert )};\ X\text{ isotropic and log-concave on }\mathbb{R}^{n}\Bigr )}.}$$

In particular, we show that if the thin-shell conjecture σ n  = O(1) holds, then n 1∕4 can be replaced by log(n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.

Keywords

  • Random Vector
  • Convex Body
  • Symmetric Convex Body
  • Uniform Probability Measure
  • Continuous Probability Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

The authors wish to thank Bo’az Klartag for a fruitful discussion and Bernard Maurey for allowing them to use an unpublished result of his.

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Correspondence to Ronen Eldan .

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Eldan, R., Lehec, J. (2014). Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates. 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_9

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