I have a question about the derivative of a distance function.
Let $D \subset \mathbb{R}^{d}$ be a connected and unbounded open subset with smooth boundary. $B(z,r)$ denotes the open ball of radius $r>0$ centered at $z \in \bar{D}$. We define the following distance function $F$ on $\mathbb{R}^{d}$:
\begin{equation*}
F:x \mapsto d(x,\partial D \cap B(z,r)).
\end{equation*}
This function is differentiable in a.e. sense since it is Lipschitz continuous (Rademacher's theorem).
Can we show that the following estimate holds?
\begin{equation*}
\text{ess inf}_{x \in \mathbb{R}^{d}} |\nabla F(x)|>0
\end{equation*}
More weakly, can we show that the following?
\begin{equation*}
|\nabla F(x)|>0 \text{ a.e.}
\end{equation*}
If you know related results, please let me know.
Best Answer
Let $K$ be a compact set with smooth boundary. The distance function has gradient 1 everywhere where the gradient exists. The gradient exists in any $x$ there exists a unique $y \in \partial K$ boundary point minimizing the distance $d(x,y) = d(K,x)$. The proof is simple. Take the normal at $y$ and map a neighbourhood.