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

TL;DR

This paper proves that weak convergence of $L_p$ Gaussian surface area measures leads to convergence of convex bodies and shows the solution to the Gaussian Minkowski problem varies continuously with $p$, advancing understanding in convex geometry.

Contribution

It establishes the continuity of solutions to the $L_p$ Gaussian Minkowski problem with respect to the parameter $p$, linking measure convergence to geometric convergence.

Findings

Weak convergence of measures implies Hausdorff convergence of convex bodies for $p \\geq 1$

The solution to the Gaussian Minkowski problem varies continuously with $p$

Provides a new link between measure convergence and geometric convergence in Gaussian 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 $p\geq 1$. Moreover, this paper obtains the solution to the Guassian Minkowski problem is continuous with respect to $p$.