Paouris, Grigoris et Valettas, Petros, A Gaussian small deviation inequality for convex functions, Annals of probability (Online) , 46(3), 2018, pp. 1441-1454
Let Z be an n-dimensional Gaussian vector and let f:Rn.R be a convex function. We prove that P(f(Z).Ef(Z).tVarf(Z)........).exp(.ct2), for all t>1 where c>0 is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.