Show that function is a diffeomorphism Consider two non-empty, bounded and convex domains $A \subset B \subset \mathbb{R}^d$ with $C^2$-boundaries.
Define a function $\phi:\partial A \to \partial B$ by $x\mapsto y$ such that $y-x=\lambda_x\vec{n}_A(x)$ for some $\lambda_x >0$ where $\vec{n}_A(x)$ is the outer normal of $\partial A$ at $x$. So roughly speaking this function maps any boundary point of $A$ to its "orthogonal counterpart" on the boundary of $B$. I expect that this function is homeomorphic (this should follow from the fact that any boundary of a convex set is homeomorphic to the unit disk) and I also guess that it is a diffeomorphism due to the $C^2$-boundaries but I don't know how to show this. In particular I am interested in a explicit expression for $|\det (D\phi(x))|$ as I want to use the integration by substitution formula. Can anyone help me with these two problems?
I appreciate any help.
 A: The proof should follow that of the $\epsilon$-neighbourhood theorem in Guillemin and Pollack: c.f. the following excerpt of the theorem statement, Page 69-70:
G&P work with smooth manifolds, though in this case $C^2$ will suffice; by diffeomorphism I take it you mean $C^1$-diffeomorphism, i.e. $C^1$ with $C^1$ inverse. I'm not sure if you are familiar with differential topology, and I haven't checked the details myself, but the idea is to consider the map $h: N(\partial A) \to \mathbb R^d$ given by 
\begin{equation*}
h(x, n) = x + n
\end{equation*}
where $N(\partial A)$ denotes the normal bundle of $\partial A$. This map is $C^2$ and by convexity $h \pitchfork \partial B$, that is, $h$ is transverse to $\partial B$. In particular $h^{-1} (\partial B)$ is a codimension 1 submanifold of $\mathbb R^d$. The restriction of $h$ to $h^{-1} (\partial B)$ should still be a submersion, i.e. the Jacobian is surjective, and $h$ maps onto $\partial B$, so arguing by rank nullity $h$ is a diffeomorphism from $h^{-1} (\partial B) \to \partial B$. The projection map $\pi: h^{-1} (\partial B) \to \partial A$ given by $(x, b) \mapsto x$ also furnishes a diffeomorphism by similar reasoning. 
Then we are done, since $h \circ \pi^{-1}: \partial A \to \partial B$ is exactly the map we are concerned about. This might seem like a bunch of hocus pocus but I think enough of the proof idea should be there for you to fill in the proof, even if a few details are wrong. 
