Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

TL;DR

This paper proves that convex bodies are uniquely minimal fillings of their boundary metrics among integral current spaces and establishes their Lipschitz-volume rigidity, extending known smooth category results to a broader setting.

Contribution

It introduces the minimal filling property of convex bodies among integral current spaces and demonstrates Lipschitz-volume rigidity in this context, answering related open questions.

Findings

Convex bodies are unique minimal fillings among integral current spaces.

Convex bodies exhibit Lipschitz-volume rigidity in the integral current space setting.

Addresses intrinsic flat convergence of minimizing sequences for the Plateau problem.

Abstract

In this paper we consider metric fillings of convex bodies. We show that convex bodies $C\subset \mathbb{R}^n$ are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As a further application of this result, we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.