Smooth solutions to the Gauss image problem

TL;DR

This paper introduces a geometric flow approach to solve the Gauss image problem, establishing the existence of smooth solutions and extending classical convex geometry problems.

Contribution

It presents a novel flow-based method to prove the existence of smooth solutions to the Gauss image problem, generalizing the Aleksandrov problem.

Findings

Existence of smooth solutions established

Flow method successfully applied to the problem

Generalization of Aleksandrov problem achieved

Abstract

In this paper we study the the Gauss image problem, which is a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.