Let $B$ denote the closed unit ball in $\mathbb{R}^d$ and $X \subset B$ be diffeomorphic to $B$. Let $\varphi: B \to X$ denote a diffeomorphism from $B$ to $X$.
I was looking for conditions (at least sufficient, if not necessary) on $X$ which ensures that there exists a $\varphi$ such that $0 < c \leq|\det J_{\varphi}(x)| \leq C$ holds for all $x$ for some $c, C$ independent of $x$. Here $J_{\varphi}$ denotes the Jacobian of the function $\varphi$.
In other words, I want $X$ to be "sufficiently regular" to ensure that when I map $B$ to $X$, the volume element does not change "too much". It looks like $X$ should satisfy some sort of measure density condition but I am not sure how to show that is sufficient (and/or necessary).
Any leads or references would be highly appreciated.