I've been reading the chapter about currents in Simon's Geometric Measure Theory (see e.g. http://web.stanford.edu/class/math285/ts-gmt.pdf). In section 2 he defines the Mass of an $n$-current $T$ by $$M(t):=\sup_{|w|\leq 1, w\in D^n}T(w)$$ and in Remark 2.6 he mentions that there is some freedom in defining this Mass. If instead of using the euclidean norm $|\cdot|$ we would use a Norm $\|\cdot\|$ on $\Lambda^n \mathbb{R}^p$ with the following properties $$(1)\ \xi\in\Lambda_n\mathbb{R}^p\mbox{ simple }\Rightarrow \langle \omega,\xi\rangle \leq \|\omega\||\xi| \mbox{ for every }\omega\in \Lambda^n\mathbb{R}^p$$ $$(2) \mbox{ For every } \xi\in\Lambda_n\mathbb{R}^p \mbox{ exists } \omega\in \Lambda^n\mathbb{R}^p \mbox{ with equality in }(1).$$ Let us define the mass of $T$ with respect to this new norm by $$M_{\|\cdot\|}(T)=\sup_{\|w\|\leq 1, w\in D^n}T(w).$$
Can we expect some kind of estimate between these two masses? What I mean is one of the following correct? $$M_{\|\cdot\|}(T)\leq M(T)$$ $$M(T)\leq M_{\|\cdot\|}(T)$$
I think something like this can be expected since Simon also gives an example for such a norm $\|\cdot\|$ which is called the comass norm and is given by $$\|\omega\|:=\sup_{|\xi|\leq 1, \xi \mbox{ simple}}\langle \omega,\xi\rangle.$$ This norm satisfies $$\|\omega\|\leq \sup_{|\xi|\leq 1,\xi \mbox{ simple}}|\omega||\xi|\leq |\omega|,$$ which in turn gives one of the above inequalities. So the question is can we expect one of these inequalities to hold for general norms satisfying (1) and (2)?