A partial differential equation for the strictly quasiconvex envelope

Abstract

In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for $\e$-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite difference approximations, comparing the QC envelope and the two regularization. Solutions of this PDE are strictly convex, and smoother than the robust-QC functions.