Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)

Abstract

We show that, for any metric probability space (M, d, μ) with a subgaussian constant σ2(μ) and any Borel measurable set A ⊂ M, we have \(\sigma ^{2}(\mu _{A}) \leq c\log \left (e/\mu (A)\right )\sigma ^{2}(\mu )\), where μA is a normalized restriction of μ to the set A and c is a universal constant. As a consequence, we deduce concentration inequalities for non-Lipschitz functions.

2010 Mathematics Subject Classification.

Primary 60Gxx 

Notes

Acknowledgements

The authors gratefully acknowledge the support and hospitality of the Institute for Mathematics and its Applications, and the University of Minnesota, Minneapolis, where much of this work was conducted. The second named author would like to acknowledge the hospitality of the Georgia Institute of Technology, Atlanta, during the period 02/8-13/2015.

We would like to thank anonymous referee for pointing out to us several relevant references. S.G. Bobkov’s research supported in part by the Humboldt Foundation, NSF and BSF grants. P. Nayar’s research supported in part by NCN grant DEC-2012/05/B/ST1/00412. P. Tetali’s research supported in part by NSF DMS-1407657.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Institute of Mathematics and its ApplicationsMinneapolisUSA
  3. 3.School of Mathematics and School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA

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