Geodesic
Deviation inequalities for convex functions motivated by the Talagrand conjecture
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 168-182

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Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on integrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure. 
@article{ZNSL_2017_457_a8,
     author = {N. Gozlan and M. Madiman and C. Roberto and P. M. Samson},
     title = {Deviation inequalities for convex functions motivated by the {Talagrand} conjecture},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {168--182},
     publisher = {mathdoc},
     volume = {457},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a8/}
}
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N. Gozlan; M. Madiman; C. Roberto; P. M. Samson. Deviation inequalities for convex functions motivated by the Talagrand conjecture. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 168-182. http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a8/