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Inner Regularization of Log-Concave Measures and Small-Ball Estimates

  • Bo’az Klartag
  • Emanuel Milman
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying small-ball information. We propose an alternative preprocessing step for making the measure seem super-Gaussian, at least up to reasonably high moments, which does not suffer from this caveat: namely, convolving the measure with a random orthogonal image of itself. As an application of this “inner-thickening”, we recover Paouris’ small-ball estimates.

Keywords

Convex Body Concentration Property Euclidean Ball Standard Gaussian Random Variable Unit Euclidean Ball 
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.

Notes

Acknowledgements

We thank Olivier Guédon and Vitali Milman for discussions. Bo’az Klartag was supported in part by the Israel Science Foundation and by a Marie Curie Reintegration Grant from the Commission of the European Communities. Emanuel Milman was supported by the Israel Science Foundation (grant no. 900/10), the German Israeli Foundation’s Young Scientist Program (grant no. I-2228-2040.6/2009), the Binational Science Foundation (grant no. 2010288), and the Taub Foundation (Landau Fellow).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Sackler Faculty of Exact ScienceTel-Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

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