On Busemann--Hausdorff densities of dimension two and of codimension two, with an application to Plateau Problem

TL;DR

This paper explores convexity properties of Busemann--Hausdorff densities in specific dimensions, establishing new cases of convexity and providing counterexamples to previous assumptions, with applications to the Plateau problem.

Contribution

It identifies a new case where Busemann--Hausdorff densities are convex and demonstrates that such convexity does not hold universally in higher dimensions.

Findings

Convexity of Busemann--Hausdorff densities in a new specific case.

Existence of non-convex Busemann--Hausdorff densities in higher dimensions.

Application to the existence of minimizing rectifiable chains in normed spaces.

Abstract

The purpose of this paper is twofold. First, we describe one (presumably) new case, in which Busemann--Hausdorff densities are convex. We apply the corresponding result to prove the existence of minimizing rectifiable chains of codimension two in complex finite dimensional normed vector spaces. Second, we prove that for each $n\geq 4$, there exists an $n$ dimensional normed space in which the corresponding two dimensional Busemann--Hausdroff density is not totally convex. This gives a negative answer to a question posed by H. Busemann and E. Strauss.