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

Abstract

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