Uniform approximation by a non-convex cone of continuous functions

Abstract

Let Π be a collection of subsets of a compact set S in a normed linear space and K be all continuous functions ƒ on S whose level sets, {s: ƒ(s) ⩽ α}, are in Π for all α. Then K is a cone which is not necessarily convex. The problem under consideration is to find a best uniform approximation to a continuous function on S from K. In this article, under certain conditions on Π, extremal best approximations are identified, a best approximation and its uniqueness are characterized, and Lipschitzian selections are determined. The results are illustrated by approximation problems. Analysis is also presented for the special case when K is a convex cone. Applications are given to normed vector lattices and the isotone approximation problem on order-intervals.