  Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and
satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u
= 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on
$\partial \Omega$. We prove that the unique solution to this Dirichlet problem
is power-concave (precisely, 3/4 concave) and it is of class $C ^1(\Omega)$. We
then investigate the overdetermined Serrin-type problem obtained by adding the
extra boundary condition $|\nabla u| = a$ on $\partial \Omega$; by using a
suitable $P$-function we prove that, if $\Omega$ satisfies the same assumptions
as above and in addition contains a ball with touches $\partial \Omega$ at two
diametral points, then the existence of a solution to this Serrin-type problem
implies that necessarily the cut locus and the high ridge of $\Omega$ coincide.
In turn, in dimension $n=2$, this entails that $\Omega$ must be a stadium-like
domain, and in particular it must be a ball in case its boundary is of class
$C^2$.
