On the \(s\)-Gaussian measure in \(\mathbb{R}^n\) (Q6614407)

In the paper under review the authors construct the \(s\)-Gauss probability space \(\mathbb{R}^n\) equipped with the \(s\)-Gaussian measure \(\gamma_n^s\), which is defined by\N\[\N\gamma_n^s(E)=\left(\dfrac{n\omega_n}{2}\left(\dfrac{2}{s}\right)^{\frac{n}{2}} \beta\left(\frac{1}{s}+1,\frac{n}{2}\right)\right)^{-1}\int_E\left(1-\frac{s|x|^2}{2}\right)^{\frac{1}{s}}_+dx\N\]\Nfor any measurable Borel subset \(E\subset\mathbb{R}^n\), where \(\omega_n\) denotes the volume of the unit ball \(B_n\). Then, the classical Ehrhard symmetrization for sets in \(\mathbb{R}^n\) is extended to a so-called \((s,k)\)-Ehrhard symmetrization, \(k\in\{1,\dots,n\}\), which is used, in particular, to obtain the \(s\)-Gaussian isoperimetric inequality in the planar case: for any Borel subset \(E\subset\mathbb{R}^2\) and all \(r\geq 0\), if \(H\) is a half-space in \(\mathbb{R}^2\) such that \(\gamma_2^s(H)=\gamma_2^s(E)\), then \(\gamma_2^s(H_r)\leq\gamma_2^s(E_r)\); here \(E_r=E+rB_n\). They also extend the classical Ehrhard inequality in terms of the standard Gaussian measure to the \(s\)-Gaussian measure in \(\mathbb{R}^1\).