Generalized Minkowski inequality via degenerate Hessian equations on exterior domains

TL;DR

This paper establishes a generalized Minkowski inequality for smooth, star-shaped domains by solving degenerate k-Hessian equations on exterior domains, advancing geometric analysis techniques.

Contribution

It introduces a novel approach using degenerate k-Hessian equations to prove Minkowski inequalities on exterior domains, extending classical geometric inequalities.

Findings

Proved a generalized Minkowski inequality for star-shaped domains.

Established solvability of degenerate k-Hessian equations on exterior domains.

Connected geometric inequalities with PDE methods.

Abstract

In this paper, we prove a generalized Minkowski inequality holds for any smooth, $(k-1)$-convex, starshaped domain $\Omega.$ Our proof relies on the solvability of the degenerate $k$-Hessian equation on the exterior domain $\mathbb R^n\setminus\Omega.$