Normal vector to a regular star shaped region Let $U\subset \mathbb{R}^N$ be a star-shaped region (with respect to the origin $0$) with $C^1$ bundary.
Can you give a detailed proof of the following fact?

if $x\in\partial U$ then for each $\varepsilon >0$ there exists $\delta >0$ such that $|y-x|<\delta$ and $y\in\bar{U}$ imply 
  $$\nu(x)\cdot\frac{y-x}{|y-x|}\le\varepsilon.$$


We say that $\Omega \subset \mathbb{R}^N$ has $C^k$ boundary if, for all $x \in \partial \Omega$, there exists a neighborhood $O$ of $x$ and a map $T:Q \to O$ which is bijective, of class $C^k$ and with inverse of class $C^k$ and such that 
$$T(Q_+) = O \cap \Omega$$
$$T(Q_0)= O \cap \partial \Omega,$$
where we have denoted 
$$Q=\{ (x',x_N) \in \mathbb{R}^{N-1}\times \mathbb{R}, |x'|< 1, |x_N|< 1\}$$
$$Q_+=\{ x \in Q: x_n >0\}$$
$$Q_0=\{ x \in Q: x_n =0\}$$
 A: It seems to me that the star-shaped assumption is unneccessary, since this is a purely local claim. To reduce confusion I'm going to rename your $x \in \partial \Omega$ to $p$, and WLOG assume that $T(0) = p$. 
Since $\nu$ is the outwards normal vector, we must have $Z=\partial T/\partial x_N|_0 \cdot\nu < 0$, since travelling in the $+x_N$ direction moves in to the interior of $Q_+$.
Applying Taylor's theorem to $T$, we get $$T(x', x_N) = p + \frac{\partial T}{\partial x_N}\Big|_0 x_N + \frac{\partial T}{\partial x'}\Big|_0 \cdot x' + o(|x',x_N|)$$ and thus $$(T(x', x_N) -
 T(0) )\cdot \nu = Z x_N + o(|x',x_N|),$$ where the $x'$ term vanished because $\nu$ is orthogonal to $\partial T/\partial x'|_0$ by definition. Thus for any $\epsilon' > 0$ we can find a $\delta_1 > 0$ such that $$x_N \ge 0, |x',x_N| < \delta_1 \implies \left(\frac{T(x',x_N) - p}{|x',x_N|}\right)\cdot \nu<\epsilon'.$$ Since $T$ has invertible derivative at $0$, we can find $\delta_2>0$ such that $$|x',x_N| < \delta_2 \implies |x',x_N| < C |T(x',x_N) - p|$$ where $C$ is a fixed constant (anything larger than the operator norm of $|DT_0^{-1}|$ will do). Putting these together we find that $$x_N \ge 0, |x',x_N| <  \min(\delta_1, \delta_2) \implies \left(\frac{T(x',x_N) - p}{|T(x',x_N) - p|} \right) \cdot \nu < \frac {\epsilon'} C.$$
Since $T : Q_+ \cup Q_0 \to \bar \Omega \cap O$ is a bijection with continuous inverse, any $y \in \bar \Omega$ sufficiently close to $p$ can be written as $T(x',x_N)$ for some $(x',x_N)$ satisfying these assumptions, so choosing $\epsilon' = C \epsilon$ completes the proof.
