A note on the convex infimum convolution inequality (Q2405181)

The authors characterize the symmetric real random variables which satisfy a convex infimum convolution inequality due to \textit{B. Maurey} [Geom. Funct. Anal. 1, No. 2, 188--197 (1991; Zbl 0756.60018)]. Let \(\mathcal{M}(h,\lambda)\) be the class of symmetric real random variables \(X\) such that \(\mathbb{P}(X\geq x+h)\leq\lambda\mathbb{P}(X\geq x)\) for all \(x\geq0\). The main result of the paper is the equivalence of the following three statements: (1) \(X\in\mathcal{M}(h,\lambda)\); (2) \(X\) satisfies a convex Poincaré inequality; and (3) there is \(C_\tau>0\) such that \[ (\mathbb{E}e^{f\square\varphi(X)})(\mathbb{E}e^{-f(X)})\leq1\, \] for all convex functions \(f:\mathbb{R}\mapsto\mathbb{R}\) with \(\inf f>-\infty\), where \(\varphi(x)=\varphi_0(x/C_\tau)\), \[ \varphi_0(x) = \begin{cases} \frac{1}{2}x^2, & |x|\leq1, \\ |x|-\frac{1}{2}, & |x|>1, \end{cases} \] and \((f\square\varphi)(x)=\inf_y\{f(y)+\varphi(x-y)\}\), the infimum convolution. This generalizes a theorem due to Maurey [loc. cit.], and also allows the authors to deduce two-level concentration for a random vector \((X_1,\ldots,X_n)\) whose entries are independent random variables in the class \(\mathcal{M}(h,\lambda)\). This extends exponential concentration results due to \textit{S. G. Bobkov} and \textit{F. Götze} [Probab. Theory Related Fields 114, No. 2, 245--277 (1999; Zbl 0940.60028)].