# Deviation inequalities for convex functions motivated by the Talagrand conjecture

## TL;DR

This paper investigates deviation inequalities for log-semiconvex functions under Gaussian measure, inspired by Talagrand's conjecture on regularization properties of certain semigroups, bridging discrete and continuous cases.

## Contribution

It introduces new deviation inequalities for convex functions motivated by Talagrand's conjecture, connecting discrete hypercube and Gaussian settings.

## Key findings

- Derived deviation inequalities for log-semiconvex functions under Gaussian measure
- Established connections between Talagrand's conjecture and regularization properties of semigroups
- Extended understanding of convex function behavior in probabilistic frameworks

## Abstract

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 in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.