Viscosity characterizations and convex analysis of quasiconvex and robustly quasiconvex functions

Abstract

A quasiconvex function is a function whose sublevel sets are convex. A function which is quasiconvex under every linear perturbation is a convex function. A robustly quasiconvex function (also called s-quasiconvex) is a function which is quasiconvex under sufficiently small linear perturbations.

The talk will characterize quasiconvex and robustly quasiconvex nonsmooth functions as viscosity solutions of certain second-order partial differential equations. Some related uniqueness results for boundary problems will be mentioned. Examples and convex-analytic properties of robustly quasiconvex functions will be presented.

This is joint work with E.N. Barron and R.R. Jensen.

Details