  Aim of this paper is to prove necessary and sufficient conditions on the
geometry of a domain $\Omega \subset \mathbb{R}^n$ in order that the
homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$
with constant source term admits a viscosity solution depending only on the
distance from $\partial \Omega$. This problem was previously addressed and
studied by Buttazzo and Kawohl. In the light of some geometrical achievements
reached in our recent paper "On the characterization of some classes of
proximally smooth sets", we revisit the results obtained by Buttazzo and Kawohl
and we prove strengthened versions of them, where any regularity assumption on
the domain and on the solution is removed. Our results require a delicate
analysis based on viscosity methods. In particular, we need to build suitable
viscosity test functions, whose construction involves a new estimate of the
distance function near singular points.
