Which domains have two-sided supporting unit spheres at every boundary point?

TL;DR

This paper establishes a precise equivalence between the geometric condition of having two-sided supporting spheres at every boundary point and the Lipschitz continuity of the outward normal vector, with extensions to infinite-dimensional spaces.

Contribution

It proves the quantitative equivalence of two key geometric regularity conditions and extends the results to infinite-dimensional $L^p$ spaces.

Findings

Domains with Lipschitz continuous outward normals have two-sided supporting spheres.

The equivalence holds with Lipschitz constant one.

Extension to infinite-dimensional $L^p$ spaces.

Abstract

We prove the quantitative equivalence of two important geometrical conditions, pertaining to the regularity of a domain $\Omega\subset\mathbb{R}^N$. These are: (i) the uniform two-sided supporting sphere condition, and (ii) the Lipschitz continuity of the outward unit normal vector. In particular, the answer to the question posed in our title is: "Those domains, whose unit normal is well defined and has Lipschitz constant one." We also offer an extension to infinitely dimensional spaces $L^p$, $p\in (1,\infty)$.