  We study the PDE $\lambda_j(D^2 u) = 0$, in $\Omega$, with $u=g$, on
$\partial \Omega$. Here $\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)$ are
the ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometric
interpretation of the viscosity solutions to the problem in terms of
convex/concave envelopes over affine spaces of dimension $j$. In one of our
main results, we give necessary and sufficient conditions on the domain so that
the problem has a continuous solution for every continuous datum $g$. Next, we
introduce a two-player zero-sum game whose values approximate solutions to this
PDE problem. In addition, we show an asymptotic mean value characterization for
the solution the the PDE.
