Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in $\R^3$

TL;DR

This paper proves convexity and rearrangement properties of solutions to certain Hessian equations in three-dimensional space, establishing Brunn-Minkowski and isoperimetric inequalities for related functionals using Minkowski addition of convex functions.

Contribution

It introduces new convexity and rearrangement results for Hessian equations in D and derives associated geometric inequalities, expanding the understanding of these equations' solutions.

Findings

Convexity of solutions to Hessian equations in D

Brunn-Minkowski inequalities for related functionals

Isoperimetric inequalities derived from the analysis

Abstract

By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.