I am studying geometric measure theory and there's a thing that always bothers me when I think about it: the definition of comass norm. It's all nice and clear, don't get me wrong: given a $m$-covector $\xi$ we define its comass norm to be
\begin{equation} ||\xi||^* = \sup_{|v| =1, \text{ simple}} |\langle \xi ,v \rangle | \end{equation}
where $v$ is a $m$-vector and $|\cdot |$ is the standard euclidean norm, i.e. if $v=v_1 \wedge \dots \wedge v_m$, then $|v|= \sqrt{det (\langle v_i , v_j \rangle })$. Then you define the mass norm for $m$-vectors as the dual of $|| \cdot ||^*$, so far so good. But why did someone think of imposing the fact that $v$ must be simple? Now, if I remember correctly in general you can associate $m$-vectors with an oriented $m$-plane in $\mathbb{R}^n$ in a canonical way only if $v$ is simple, but still I cannot see why to bother or why someone thought it was necessary to add the "simple" condition to the definition.