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

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 $L^p$, $p\in (1,\infty)$.