CONTINUITY PROPERTIES OF THE NORMAL CONE TO THE LEVEL SETS OF A QUASI-CONVEX FUNCTION

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the function x→{norm of matrix}∇f(x){norm of matrix}-1∇f(x) on the domain where it is defined is deduced from some continuity properties of the normal cone N to the level sets of the quasiconvex function f. We also prove that, under a pseudoconvexity-type condition, the normal cone N(x) to the set {x:f(x)≤f(x)} can be expressed as the convex hull of the limits of type {N(xn)}, where {xn} is a sequence converging to x and contained in a dense subset D. In particular, when f is pseudoconvex, D can be taken equal to the set of points where f is differentiable. © 1990 Plenum Publishing Corporation.