Let $c>0$ be a real positive number and let $T\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a diffeomorphism such that $\left|T\left(x\right)-T\left(y\right)\right|\geq c\left|x-y\right|$. I wish to show that for every Jordan Measurable set $E$ we have $v\left(T\left(E\right)\right)\geq c^{n}v\left(E\right)$.
In my attempt I realized that using change of variables we get $$ v\left(T\left(E\right)\right)=\int_{T\left(E\right)}1=\int_{E}\left|J_{T}\left(x\right)\right| $$ where $J_{T}$ is the Jacobian of $T$, and therefore it is sufficient to show that for every $x\in E$ we have $\left|J_{T}\left(x\right)\right|\geq c^{n}$. I couldn't see how to prove that so I was thinking about looking first on the case where $c=1$, but also there I couldn't figure how to do it. Any suggestions?
EDIT:
More I know is that for every $x,h\in\mathbb{R}^{n}$ we have $T\left(x+h\right)=T\left(x\right)+D_{T}\left(x\right)\left(h\right)+o\left(\left|h\right|\right)$ and therefore $$ \begin{align*} c\left|h\right| & \leq\left|T\left(x+h\right)-T\left(x\right)\right|=\left|D_{T}\left(x\right)\left(h\right)+o\left(\left|h\right|\right)\right|\leq\\ & \leq\left|D_{T}\left(x\right)\left(h\right)\right|+\left|o\left(\left|h\right|\right)\right|\leq\left\Vert D_{T}\left(x\right)\right\Vert \left|h\right|+o\left(\left|h\right|\right) \end{align*} $$ so $c\leq\left\Vert D_{T}\left(x\right)\right\Vert +\frac{o\left(\left|h\right|\right)}{\left|h\right|}$ and when $h\to0$ we get $\left\Vert D_{T}\left(x\right)\right\Vert \geq c$
$T(x+tv)=T(x)+dT_x(tv)+tO(t)$. We deduce that $\|{{\|T(x+tv)-T(x)\|}\over{\|tv\|}}=\|dT_x({v\over \|v\|})+O(t)\|\geq c$. This implies that $\|dT_x(v)\|\geq c\|v\|$.
Let $e$ be an eigenvalue of $dT_x$ associated to the eigenvector $u$ (I will eventually work with the complexification), we have:
$T(x+tv)=T(x)+dT_x(tv)+tO(t)=T(x)+etv+tO(t)$, we deduce that:
$\|{{\|T(x+tv)-T(x)\|}\over{\|tv\|}}=\|e({v\over \|v\|})+O(t)\|\geq c$.