The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions
Emmanuel Nicholas Barron, Rafal K. Goebel, Robert R. Jensen · Discrete and Continuous Dynamical Systems - B · 2012
Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a functionare given in terms of first-order partial differential equations. Solutions to the equationsare understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuousfunctions. A comparison principle, implying uniqueness of solutions, is shown for a relatedpartial differential equation. This equation is then used in an iterative construction ofthe quasiconvex envelope of a function. The results are then extended to robustly quasiconvexfunctions, that is, functions which are quasiconvex under small linear perturbations.