In [1], Kirchheim proved the area formula for Lipschitz maps $f\colon \mathbb{R}^n\to X$ where $X$ is an arbitrary metric space, using the notion of metric differentiability. The metric derivative of $f$ at $x$, which Kirchheim proved exists almost everywhere (Kirchheim-Rademacher theorem), is a seminorm on $\mathbb{R}^n$. To state an area formula, a Jacobian must be defined for seminorms.
If a seminorm is degenerate, that is, it vanishes on some nonzero vector, then we set its Jacobian to zero. So, suppose $\sigma$ is a norm on $\mathbb{R}^n$. Since, $\sigma$ defines a metric structure on $\mathbb{R}^n$, we can talk about the Hausdorff $n$-measure with respect to $\sigma$, which we denote by $\mathcal{H}^n_\sigma$. Jacobian $J(\sigma)$ must be defined such that \begin{equation} \mathcal{H}^n_\sigma(A)=J(\sigma)\mathcal{L}^n(A), \quad \text{for all $A\subset \mathbb{R}^n$.} \end{equation} We can argue (by its translation invariance) that $\mathcal{H}^n_\sigma$ is a constant multiple of $\mathcal{L}^n$, therefore one possible definition (from taking $A=[0,1]^n$) is $$ J(\sigma)=\mathcal{H}^n_\sigma([0,1]^n). \qquad (\text{Definition 1}) $$ However, we might not like to work with $\mathcal{H}^n_\sigma$, and in fact there is a way to avoid this. If we denote by $B_\sigma$ the closed unit ball of $\sigma$, then taking $A=B_\sigma$ in the above equality gives $$ J(\sigma)=\frac{\mathcal{H}^n_\sigma(B_\sigma)}{\mathcal{L}^n(B_\sigma)}. $$ Amazingly, the numerator is independent of $\sigma$.
Theorem ([1], Lemma 6): For any norm $\sigma$ on $\mathbb{R}^n$, the quantity $\mathcal{H}^n_\sigma(B_\sigma)$ equals $\omega_n$, the (Lebesgue) volume of the Euclidean ball.
In light of this result, the definition used in [1] is $$ J(\sigma)=\frac{\omega_n}{\mathcal{L}^n(B_\sigma)}. \qquad (\text{Definition 2}) $$ Notice that this definition only involves the Lebesgue measure. The proof of the theorem in Kirchheim uses results about the density of measures and an isodiametric inequality for (finite dimensional) normed spaces.
Question: Are you aware of any proof of the theorem above other than the one given by Kirchheim.
I will be interested in special cases and partial results as well. Thanks!
[1] Rectifiable Metric spaces: Local Structure and Regularity of the Hausdorff Measure, Bernd Kirchheim, Proceedings of the American Mathematical Society, Vol. 121, No. 1 (May, 1994), pp. 113-123 (11 pages)
