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 Gδ set, an Fσ set, and d − 2 sets each of which is the
intersection of a Gδ set with an Fσ set. A sharper assertion is proved for the
three-dimensional case, and some related results are obtained for certain
infinite-dimensional situations. Section 2 describes a compact convex set in the space
Rc which has no algebraically exposed points. Both sections contain unsolved
problems.
