  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.
