Bounds on Jacobian in terms of Volume Distortion Let $V\subset U\subseteq \mathbb{R}^k$ and let $\varphi:U\to V$ be a diffeomorphism. Let $J_{\varphi}$ be its Jacobian, so $vol(V) = \int_U |\det J_{\varphi}|$. I am wondering if there are any general purpose lower bounds on $|\det J_{\varphi}(u)|$ over all $u\in U$. For example, if $\varphi$ is measure/volume-preserving, $|\det J_{\varphi}| = 1$. My intuition is that something like $$|\det J_{\varphi}|\ge \inf_{S\subset U}\frac{vol(S)}{vol(\varphi(S))}$$ should hold (with the convention that $\tfrac{0}{0} = 1$), but I'm unable to find any resources (and have not been able to furnish a proof nor counterexample -- my multivariable calculus is very rusty). Also, for my particular purpose I know that $vol(\varphi(S))\le vol(S)$ for all $S\subset U$. Any help/hints are appreciated!
EDIT: My intuition also says that an inequality of the other direction should hold as well: $$|\det J_{\varphi^{-1}}|\le \sup_{T\subset V}\frac{vol(\varphi^{-1}(T))}{vol(T)}.$$ Also, in my particular setup, $U = [0,1]^k$. An idea I have for this is to bound the Jacobian by that of a linear map that $\phi([0,s]^k) = [0,1]^k$ that scales points up by a factor $1/s$ (for $0 < s < 1$). We'd choose $s$ to be minimal such that $[0,s]^k\supset V$ or something. But I'm not sure how to make meaningful use of this train of thought.
 A: We can prove your (inverted) conjecture when $\varphi$ is $\mathcal{C}^1$ : $$|\det J_{\varphi}|\ge \inf_{S\subset U}\frac{vol(\varphi(S))}{vol(S)}$$
Let $x_0 \in U$ and $\varepsilon >0$. As $J_{\varphi}$ is continuous, there is  a neighborhood $S$ of $x$ such that $|\det J_{\varphi}| < |\det J_{\varphi}(x_0)|+\varepsilon$ on $S$. Then
$$vol(\varphi(S))= \int_S |\det J_{\varphi}| \leq \int_S (|\det J_{\varphi}(x_0)|+\varepsilon)$$
hence
$$vol(\varphi(S)) \leq vol(S)\times (|\det J_{\varphi}(x_0)|+\varepsilon)$$
which gives :
$$|\det J_{\varphi}(x_0)| \geq \dfrac{vol(\varphi(S))}{vol(S)} - \varepsilon.$$
Therefore $|\det J_{\varphi}(x_0)| \geq \inf\limits_{S \subset U}\dfrac{vol(\varphi(S))}{vol(S)} \; \; - \varepsilon$, and as it holds for all $\varepsilon >0$, we get
$$|\det J_{\varphi}(x_0)| \geq \inf\limits_{S \subset U}\dfrac{vol(\varphi(S))}{vol(S)}.$$
We can apply this inequality to the diffeomorphism $\varphi^{-1}$ if we want.
However, such bounds can't really give you a meaningful global lower bound on $|\det J_{\varphi}|$. For exemple, for $a>0$, let $\varphi_a : [0,2] \to [0,1]$ defined by $\varphi_a(x)= \dfrac{(x+a)^2-a^2}{(2+a)^2-a^2}$. Then $\varphi_a'(0) = \dfrac{2a}{(2+a)^2-a^2} =  \dfrac{a}{2+a}$. Thus, by taking $a$ arbitrarily small, we can find a diffeomorphism $\varphi_a$ with $|\det J_{\varphi}(0)|$ as small as we want.
