On the tightness of Gaussian concentration for convex functions

@article{Valettas2019OnTT,
  title={On the tightness of Gaussian concentration for convex functions},
  author={Petros Valettas},
  journal={Journal d'Analyse Math{\'e}matique},
  year={2019}
}
  • P. Valettas
  • Published 28 June 2017
  • Mathematics
  • Journal d'Analyse Mathématique
The concentration of measure phenomenon in Gauss' space states that every $L$-Lipschitz map $f$ on $\mathbb R^n$ satisfies \[ \gamma_{n} \left(\{ x : | f(x) - M_{f} | \geqslant t \} \right) \leqslant 2 e^{ - \frac{t^2}{ 2L^2} }, \quad t>0, \] where $\gamma_{n} $ is the standard Gaussian measure on $\mathbb R^{n}$ and $M_{f}$ 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… 
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