Exposed points of convex sets

Abstract

The two sections of this note are unrelated, except that both are concerned with the exposed points of a compact convex subset K of a locally convex space E. In 1 it is proved that if K is of finite dimension d, then the set of all its exposed points can be expressed as the union of a Gs set, an Fa-set, and d -2 sets each of which is the intersection of a Gs set with an Fa-set. A sharper assertion is proved for the threedimensional case, and some related results are obtained for certain infinite-dimensional situations. Section 2 describes a compact convex set in the space $ c which has no algebraically exposed points. Both sections contain unsolved problems.