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 1, 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.
edit 2 : As pointed out in the answers to the question Pietro Majer links to, the question of 1.8.4 is addressed in Frank Morgan's article "The exterior algebra $\Lambda^k({\mathbb R}^n)$ and area minimization" (Linear Algebra and its Applications Volume 66, April 1985, Pages 1–28) where certain cases (by dimension) are proved.
