It is well-known, that given a linear transformation $f \colon \mathbb R^n \rightarrow \mathbb R^m$, where $m \ge n$, the $m$-dimensional volume of an image of any measurable subset $S \subseteq \mathbb R^n$ under the transformation can be expressed as: $$Volume(f(S)) = \sqrt{\det(A^TA)} Volume(S),$$ where $A \in \mathbb R^{m \times n}$ is a matrix of the linear transform.

If $m < n$, then this relation no longer holds (the mapping is not invertible and $\det(A^TA) = 0$). Is there any characterization of a volume change under linear transformation in this case? I am specifically interested in the case, when $S$ is a unit $L_1$-ball in $\mathbb R^n$.