Continuity of the solution to the $L_p$ Minkowski problem in Gaussian probability space

Abstract

In this paper, it is proved that the weak convergence of the $L_p$ Guassian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for $p\ge 1$. Moreover, this paper obtains the solution to the Guassian Minkowski problem is continuous with respect to $p$.