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.
"Centroid bodies and the convexity of area functionals." J. Differential Geom. 98 (3) 357 - 373, November 2014. https://doi.org/10.4310/jdg/1406552275