Gauss curvature flow to the $L_p$-Gaussian chord Minkowski problem

TL;DR

This paper introduces new Minkowski problems related to Gaussian chord measures and solves them for smooth even solutions using a Gauss curvature flow method.

Contribution

It formulates the $L_p$-Gaussian chord Minkowski problem and the log-Gaussian chord Minkowski problem, providing the first smooth solutions via curvature flow techniques.

Findings

Established existence of smooth even solutions for the $L_p$-Gaussian chord Minkowski problem.

Proposed a novel approach using Gauss curvature flow to solve these Minkowski problems.

Abstract

Recently, Huang and Qin \cite{HY01} introduced the Gaussian chord measure and $L_p$-Gaussian chord measure by variational methods. Meanwhile, they posed Gaussian chord Minkowski problem for $p=1$ and used variational methods to obtain an origin-symmetric normalized measure solution for the Gaussian chord Minkowski problem. The smooth solution, up to now, to the $L_p$-Gaussian chord Minkowski problem is still open. Motivated by the forgoing works by Huang and Qin in \cite{HY01}, we propose in the present paper the $L_p(p>0)$-Gaussian chord Minkowski problem and log-Gaussian chord Minkowski problem, and obtain the smooth even solutions to these two types of problems by the method of a Gauss curvature flow.