In Federer's book "Geometric measure theory", he says in section 1.8.3 (where $\|\cdot\|$ is the comass norm):

"Very little appears to be known about the structure of the convex sets $\wedge^m(\mathbb{R}^n)^\ast\cap\{\phi:\|\phi\|\leq 1\}$. What are their extreme points?"

In section 1.8.4 he says:

"Suppose $S$ and $T$ are mutually orthogonal subspaces of an inner product space $V$, $s:S\to V$ and $t:T\to V$ are the inclusion maps, $\xi\in\operatorname{im}\wedge_ps$ and $\eta\in\operatorname{im}\wedge_qt$. The equation $\|\xi\wedge\eta\|=\|\xi\|\cdot\|\eta\|$ holds if either $\xi$ or $\eta$ is simple. [...] I do not know whether the above equation holds in case neither $\xi$ nor $\eta$ is simple."

Are these matters understood by now?

edit, for completeness: given a finite-dimensional real inner product space $V$, the comass of $\phi\in\wedge^mV^\ast$ is defined by
$$\|\phi\|=\sup\Big\{\phi(\xi):\text{decomposable }\xi\in \wedge^mV\text{ with }|\xi|\leq 1\Big\}$$
and the mass of $\xi\in\wedge^mV$ is defined by
$$\|\xi\|=\sup\Big\{\phi(\xi):\phi\in\wedge^mV^\ast\text{ with }\|\phi\|\leq 1\Big\}.$$
Here $|\cdot|$ is the standard norm on $\wedge^mV$, defined by an orthonormal basis $e_{i_1}\wedge\cdots\wedge e_{i_k}$. It seems to me that the mass and comass are essentially analogous to (and perhaps the inspiration for) the Gromov norm on homology.