The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions

Emmanuel N. Barron, Rafal Goebel, Robert R. Jensen
2012 Discrete and continuous dynamical systems. Series B  
Dedicated to Avner Friedman on his 80th birthday. Abstract. Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then
more » ... in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations. 2000 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.
doi:10.3934/dcdsb.2012.17.1693 fatcat:dduwfjwlzbe3bax3hiipbn6owa