# Signed distance function and level set

For $$\phi\in C^1(\mathbb{R}^N)$$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $$\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \emptyset$$ consider the signed distance function:

$$d_{\phi}:\mathbb{R}^N\to\mathbb{R},\ d_{\phi}(x)=\begin{cases} -\text{dist}(x,\partial\omega_{\phi}),\ x\in\omega_{\phi} \\ 0,\ x\in\partial\omega_{\phi} \\ \text{dist}(x,\partial\omega_{\phi}),\ x\in\mathbb{R}^N\setminus\overline{\omega}_{\phi}\end{cases}$$

I have some technical questions that seem to be true about $$d_{\phi}$$.

Let's take a point $$x_0\in\mathbb{R}^N$$ with a unique $$y(0)\in\partial\omega_{\phi}$$ such that: $$|x_0-y|=\text{dist}(x_0,\partial\omega_{\phi})$$ and any $$\psi\in C^1(\mathbb{R}^N)$$ with $$|\psi(x)|.

For $$|\theta|$$ sufficiently small we associate to $$x_0$$ a set denoted $$Y(\theta)$$ such that $$\forall y\in Y(\theta)\subseteq\mathbb{R}^N$$ we have $$\phi(y)+\theta\psi(y)=0$$ and:

$$d_{\phi+\theta\psi}(x_0)=|x_0-y|.$$

How can we prove that:

(1) There is a selection for the multifunction $$Y:[0;\theta_0)\rightrightarrows \phi^{-1}(0)$$ denoted $$y=y(\theta)$$ which is continuous and differentiable for $$\theta\in [0,\theta_0)$$.

Note that we cannot apply here Michael's Selection Theorem because in general $$Y(\theta)$$ is not a convex set. (https://en.wikipedia.org/wiki/Michael_selection_theorem)

(2) $$\lim\limits_{\theta\to 0} d_{\phi+\theta\psi}(x_0)=d_{\phi}(x_0)$$

(3) Exists $$\lim\limits_{\theta\to 0}\dfrac{d_{\phi+\theta\psi}(x_0)-d_{\phi}(x_0)}{\theta}$$

I tried a lot to prove the continuity but I did not succeed.

It is well known (from KKT conditions) that $$y(0)$$ and $$y(\theta)$$ satisfy:

$$\begin{cases} \phi(y(0))=0,\ \phi(y(\theta))+\theta\psi(y(\theta))=0 \\ \lambda\nabla \phi (y(0))=y(0)-x_0,\ \lambda_{\theta}\Big (\nabla\phi(y(\theta))+\theta\psi(y(\theta))\Big )=y(\theta)-x_0\end{cases}$$

I proved easily that for $$\theta$$ sufficiently small $$\nabla\phi+\theta\nabla\psi\neq 0$$ on $$(\phi+\theta\psi)^{-1}(0)\neq \emptyset$$. Also note that if we prove $$(1)$$ then the answer to $$(2)$$ follows immediately and $$(3)$$ can be computed from KKT conditions given the result: $$-\dfrac{\psi(y(0))}{|\nabla\phi(y(0))|}$$.

Maybe the following sources will help:

Set of points with a unique closest point in a compact set

Concavity near the boundary of the distance function

Finally, are there some references that treats the signed distance function with the level set method (not with a shape derivative approach, but a functional approach)?

Your assertions are wrong. Assume that in a neighboorhood of $$(0,0)$$ we have $$\phi(x)=(x_1-1)^2-1+x_2^2-x_2^4\\ \psi(x)=x_2^2$$ Then for $$x_0=(1,0)$$ the unique minimum is $$y_0=(0,0)$$. However for $$\theta>0$$ there are multiple minima regardless of how small $$\theta$$ is.
• Nice example! However we can take a selection of that multifunction that associates to $x_0$ the points on $\phi^{-1}(0)$ which realizes the minimum distance between $x_0$ and $\phi^{-1}(0)$ that is both continuous and differentiable w.r.t. $\theta$ (in your example we could take $y>0$). I don't think that all my assertions are wrong. Somehow I could calculate the limit in $(4)$ being $-\dfrac{\psi(y(0))}{|\nabla\phi(y(0))|}$, but the details are a bit messy (even on the continuity condition) which is true in general. Dec 31 '20 at 17:20