Extremal Elements Of The Convex Cone An Of Functions

Abstract

Let Ai be the set of nonnegative real functions f on [0, 1] such that (formula presented) [0, 1], and let An, n \u3e 1, be the set of functions belonging to An-1 such that (formula presented). Since the sum of two functions in An belongs to An and since a nonnegative real multiple of an An function is an An function, the set of An functions forms a convex cone. It is the purpose of this paper to give the extremal elements (i.e., the generators of extreme rays) of this cone, to prove that they form a closed set in a compact convex set that does not contain the origin but meets every ray of the cone, and to show that for the functions of the cone an integral representation in terms of extremal elements is possible. The intersection of the An cones is the class of functions alternating of order∞. Thus, the set of these functions, which will be denoted by A∞ forms a convex cone also. The extremal elements for the convex cone A∞ are given too. © 1970 by Pacific Journal of Mathematics