I am reading the paper [Ricci Curvature and Volume Convergence](https://www.jstor.org/stable/2951841?seq=1) written by Professor Colding. In section 2, they define Lipschitz functions $b\_j^+:M\to\mathbb R$ with $|\nabla b\_j^+|=1$ and set $$\Phi=(b\_1^+,...,b\_n^+):B\_1(p)\to B\_{1+\epsilon}(0)\subset\mathbb R^n.$$ Then, why do we have $\text{Vol}(\Phi(B\_1(p)))\leq\text{Vol}(B\_1(p))$?



How can I check this volume comparison?
By approximation, we may assume that the map $\Phi$ is continuously differentiable. Then the absolute value of the Jacobian determinant of $\Phi$ is the volume of a parallelepiped with edge lengths $1$ and hence with volume $\le1$. So, the absolute value of the Jacobian determinant of $\Phi$ is $\le1$, and the volume comparison follows.


