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arXiv · 2209.12545

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

Giuliano Basso·Paul Creutz·Elefterios Soultanis
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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.

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Giuliano Basso, Paul Creutz, Elefterios Soultanis. 2022-09-26. Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces. https://doi.org/10.1515/crelle-2023-0076

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