  We introduce a new volume definition on normed vector spaces. We show that
the induced $k$-area functionals are convex for all $k$. In the particular case
$k=2$, our theorem implies that Busemann's 2-volume density is convex, which
was recently shown by Burago-Ivanov. We also show how the new volume definition
is related to the centroid body and prove some affine isoperimetric
inequalities.
