  We give a complete characterization, as "stadium-like domains", of convex
subsets $\Omega$ of $\mathbb{R}^n$ where a solution exists to Serrin-type
overdetermined boundary value problems in which the operator is either the
infinity Laplacian or its normalized version. In case of the not-normalized
operator, our results extend those obtained in a previous work, where the
problem was solved under some geometrical restrictions on $\Omega$. In case of
the normalized operator, we also show that stadium-like domains are precisely
the unique convex sets in $\mathbb{R}^n$ where the solution to a Dirichlet
problem is of class $C^{1,1} (\Omega)$.
