The Largest Eigenvalue of a Convex Function, Duality, and a Theorem of Slodkowski

Abstract

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 Dirichlet problem. Second, using the Legendre–Fenchel transform we derive a dual characterization of this largest eigenvalue in terms of convexity of the conjugate function. This dual characterization offers further insight into the nature of this largest eigenvalue and allows for an alternative proof of a necessary bound for the theorem.