Derivative of distance function to a closed, rectifiable set

Question

Let $\Gamma \subset \mathbf{R}^d$ be a closed, countably $n$-rectifiable set. Is there any reasonable way to write the derivatives $$ \frac{\partial}{\partial x_i} \mathrm{dist}\, (x,\Gamma) $$ for $x \notin \Gamma$?

It is Lipschitz and therefore differentiable almost everywhere. And since $\Gamma$ has tangent planes almost everywhere, it of course has a normal space at almost every point too, so you might hope/guess that for almost every point in some neighbourhood of $\Gamma$, there is a reasonable expression for these derivatives. Is anything like that true?

Answer 1

The distance $f: x \mapsto \mathsf{dist}(x,\Gamma)$ to a closed set $\Gamma$ in $\mathbb{R}^n$ is differentiable in $x \notin \Gamma$ iff the nearest point projection is unique; denote this by $x_\Gamma$. In $\mathbb{R}^n$ the derivative at those points is given by $(x-x_\Gamma)/\| x-x_\Gamma\|$. Note that points in $\Gamma$ are in general not points of differentiability (points in the interior of $\Gamma$ would be, but the derivative is obviously zero).

Note that we don't assume anything on $\Gamma$ except closedness which is needed to show that the set of nearest points to $x$ is compact.

The statement also holds for smooth Riemannian/Finsler manifolds if one replaces nearest point projection with "unique geodesic of length $d(x,\Gamma)$ connecting $x$ and $\Gamma$.

As I couldn't find a quick reference to this fact here a small proof:

If $f$ is differentiable at $x$ then $f(y)=f(x)+v\cdot(y-x)+o(\|x-y\|)$ for some $v$. Picking $y=(1-\epsilon)x+\epsilon z$ for a point $z\in\Gamma$ with $d(x,z)=f(x)$ it holds $f(y)=(1-\epsilon)f(x)$. But then $v=(x-z)/\|x-z\|$. In particular, the nearest point projection must be unique.

Let $(x_{n},y_{n})$ be two sequences with $x_{n}\to x$ and $y_{n}\in\Gamma$ such that $d(x_{n},y_{n})=f(x_{n})$. W.l.o.g. also assume $\frac{x_{n}-x}{\|x_{n}-x\|}\to v\in\partial B_{1}(0)$. If the nearest point projection at $x$ is unique then $y_{n}\to x_{\Gamma}$.

Now observe by convexity of $d_{z}:x\mapsto\|x-z\|$ one has \begin{align*} \|y-z\|-\|x-z\| & \ge\nabla d_{z}(x)\cdot(y-x)\\ & =\frac{(x-z)\cdot(y-x)}{\|x-z\|}. \end{align*} Thus \begin{align*} \frac{(x-y_{n})\cdot(x_{n}-x)}{\|x-y_{n}\|\|x_{n}-x\|} & \le\frac{\|x_{n}-y_{n}\|-\|x-y_{n}\|}{\|x_{n}-x\|}\\ & \le\frac{f(x_{n})-f(x)}{\|x_{n}-x\|}\\ & \le\frac{\|x_{n}-x_{\Gamma}\|-\|x-x_{\Gamma}\|}{\|x_{n}-x\|}.\\ \end{align*} Taking the limit we see that the left hand side converges to $\frac{(x-x_{\Gamma})}{\|x-x_{\Gamma}\|}\cdot v$ which is nothing but the derivative of $d_{x_\Gamma}$ at $x$ in direction $v$, i.e. the limit of the right hand side. Hence $f$ is differentiable in $x$.