  First, we provide an exposition of a theorem due to Slodkowski regarding the
largest "eigenvalue" of a convex function. In his work on the Dirichlet
problem, Slodkowski introduces a generalized second-order derivative which for
$C^2$ functions corresponds to the largest eigenvalue of the Hessian. The
theorem allows one to extend an a.e lower bound on this largest "eigenvalue" to
a bound holding everywhere. Via the Dirichlet duality theory of Harvey and
Lawson, this result has been key to recent progress on the fully non-linear,
elliptic Dirchlet problem. Second, we give a dual interpretation of this
largest eigenvalue using the Legendre-Fenchel transform, and use this dual
perspective to provide an alternative proof to an important step in the proof
of the theorem.
