The convexity of a domain and the superharmonicity of the signed distance function

Abstract:

Let $D$ be a domain in ${{\mathbf {R}}^N}$ with nonempty boundary $\partial D$ and let $u$ be the signed distance function from $\partial D$, i.e. $u = \pm$ dist according as we are in or outside $\overline D$. We prove that, for any $N \geqslant 2,u$ is superharmonic in ${{\mathbf {R}}^N}$ if and only if $D$ is convex. When $N = 2$, this criterion requires the superharmonicity of $u$ in $D$ only.