On the tightness of Gaussian concentration for convex functions

  • Petros ValettasEmail author


The concentration of measure phenomenon in Gauss’ space states that every L-Lipschitz map f on ℝn satisfies
$${\gamma _n}\left({\left\{{x:| {f(x) - {M_f}|\,\geqslant t} } \right\}} \right)\,\leqslant 2{e^{- {{{t^2}} \over {2{L^2}}},}}\quad t > 0,$$
where γn is the standard Gaussian measure on ℝn and Mf is a median of f. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when f is additionally assumed to be convex. In particular, we show that if the variance Var(f) (with respect to γn) satisfies \(\alpha L\leqslant \sqrt {{\rm{Var(}}f{\rm{)}}} \) for some 0 < α ⩽ 1, then
$${\gamma _n}\left({\left\{{x:\left| {f(x) - {M_f}} \right|\geqslant t} \right\}} \right)\,\geqslant \,c{e^{- C{{{t^2}} \over {{L^2}}}}},\quad t > 0,$$
where c, C > 0 are constants depending only on α.


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The author would like to thank Grigoris Paouris for posing him the question about the tightness of the concentration and for many fruitful discussions. He would also like to thank Peter Pivovarov for useful advice and comments, Ramon van Handel and Emanuel Milman for valuable remarks. Thanks also go to the anonymous referee whose helpful comments improved the exposition of this note.


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Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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