Given a metric space $(\mathbb{R}^n,d)$, a map $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$, and a compact set $\Sigma\subset \mathbb{R}^n$ ,suppose that $f$ is Lipschitz with constant $L$, i.e. $d(f(x),f(y)) \leq Ld(x,y)$ and invertible. Is it possible to relate the following two quantities:
- the volume of the set $\Sigma$, i.e. $q_1:=\int_{\Sigma} dV$,
- the volume of the set $f(\Sigma):=\{x \ |\ f^{-1}(x)\in\Sigma\}$, i.e. $q_2:=\int_{f(\Sigma)} dV$
I would like to obtain a relation like "$\exists k$ s.t. $q_2\leq k q_1$" and especially how $k$ depends on $L$ (and $d$ ?). Feel free to throw in additional assumptions, if the one I gave are insufficient.
Could you help me or point me to a reference that could have some insights? Thank you!
Edit
My take on this: I suspect that for a general distance $d$ my request is too difficult to derive. However, I think that if the distance is defined by a norm, i.e. $d(x,y):= ||x-y||$, then things could be easier, and I say this for the following reason:
Thinking about the classic argument given for the change of variable of 2-dimensional integrals, suppose that $\Sigma$ is an infinitesimal square given by the following vertexes $$\begin{bmatrix}u\\v\end{bmatrix},\begin{bmatrix}u+du\\v\end{bmatrix}, \begin{bmatrix}u\\v+dv\end{bmatrix}$$
transformed by $f(u,v):=[f_1(u,v),f_2(u,v)]$, modulus a translation, into the parallelogram given by $$\begin{bmatrix}f_1(u,v)\\f_2(u,v)\end{bmatrix},\begin{bmatrix}f_1(u+du,v)\\f_2(u+du,v)\end{bmatrix}, \begin{bmatrix}f_1(u,v+dv)\\f_2(u,v+dv)\end{bmatrix}.$$
To compute the area of the new shape $f(\Sigma)$, as a first approximation, we can consider the vectors
$$r_1 = \begin{bmatrix}f_1(u+du,v)-f_1(u,v)\\f_2(u+du,v)-f_1(u,v)\end{bmatrix}, r_2 = \begin{bmatrix}f_1(u,v+dv)-f_1(u,v)\\f_2(u,v+dv)-f_1(u,v)\end{bmatrix}$$
and obtain that $q_2=||r_1||_2^2||r_2||_2^2-<r_1,r_2>^2\leq||r_1||_2^2||r_2||_2^2$. If additionally, we know that the Lipschitz constant with respect to the 2-norm is $L$ then we obtain $$q_2\leq||r_1||_2^2||r_2||_2^2\leq LduLdv=L^2q_1$$ Then, since there are known inequalities between $||\cdot||_p$ and $||\cdot||_q$ I can use those to get $||r_1||_2^2\leq c||r_1||_p^2$ to derive the inequality I need for Lipschitz maps w.r.t. to any p-norm.
Does anyone spot a more general solution, maybe without the need for a norm-induced metric?