On the s-Gaussian measure in Rn

We construct the s-Gauss probability space by introducing the s-Gaussian density function in $R^n$ for $s\ge 0$, a generalization of the classic Gaussian density function. Based on the s-Gaussian density function, we propose the $(s,k)$-Ehrhard symmetrization which is an extension of the traditional Ehrhard symmetrization for sets in $R^n$. In particular, we establish the s-Gaussian isoperimetric inequality with respect to s-Gaussian measure in $R^2$. Furthermore, we propose and prove the s-Ehrhard-Borell inequalities for $s>0$ when one of the two sets is a Borel set whilst the other being a convex set as well as the case when two sets are convex in $R^1$ with different methods.