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The purpose of this paper is twofold. First, we describe one (presumably) newcase, in which Busemann--Hausdorff densities are convex. We apply thecorresponding result to prove the existence of minimizing rectifiable chains ofcodimension two in complex finite dimensional normed vector spaces. Second, weprove that for each $n\geq 4$, there exists an $n$ dimensional normed space inwhich the corresponding two dimensional Busemann--Hausdroff density is nottotally convex. This gives a negative answer to a question posed by H. Busemannand E. Strauss.

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