2 research outputs found

### The Extremal Structure Of Locally Compact Convex Sets

Let X be a locally compact closed convex subset of a locally convex Hausdorff topological linear space E. Then every exposed point of X is strongly exposed. The definitions of denting (strongly extreme) ray and strongly exposed ray are given for convex subsets of E. If X does not contain a line, then every extreme ray is strongly extreme and every exposed ray is strongly exposed. An example is given to show that the hypothesis that X be locally compact is necessary in both cases. Â© 1976 Pacific Journal of Mathematics. All rights reserved

### Extremal Elements Of The Convex Cone An Of Functions

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 orderâˆž. Thus, the set of these functions, which will be denoted by Aâˆž forms a convex cone also. The extremal elements for the convex cone Aâˆž are given too. Â© 1970 by Pacific Journal of Mathematics