As follows $\mathcal{H}^n$ denotes the $n-$dimensional Hausdorff measure and $\omega_n$ denotes the volume of the unit ball on $\mathbb R^n$.
In the article Rectiable sets in metric and Banach spaces published by Kirchhein and Ambrosio, for a seminorm $n$ on $\mathbb R^n$. They define the Jacobian of $s$ by \begin{equation*} J(s):=\frac{\omega_n }{\mathcal{H}^n(\{ x \in \mathbb R^n : s(x) \leq 1 \})} \end{equation*} In the same way Kirchhein in the article RECTIFIABLE METRIC SPACES:LOCAL STRUCTURE AND REGULARITY OF THE HAUSDORFF MEASURE defines \begin{equation*} J(s)=\frac{n \omega_n }{\int_{\mathbb{S}^{n-1}}[s(x)]^{-n}d\mathcal{H}^{n-1}(x)} \end{equation*} This expression is the "same" as the first because in the quotient we integrate a radial function. I have tried to understand why these definitions are made in this way, for example in the classical case when we have a linear function $L: \mathbb R^n \to \mathbb R^m$ and both $\mathbb R^n$, $\mathbb R^m$ are equipped with the euclidean norm, we can define the Jacobian of $L$ by \begin{equation*} J(L)=\frac{\mathcal{H}^n(L(B(0,1))}{\omega_n} \end{equation*}
I try to found a connection but i cant see these, any help i will be very grateful.