ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: Grza̧ślewicz, Ryszard, Wrocław, Poland Quasi-interior points for convex sets For a convex set Q in a topological linear space E space we put

Q^{⋆} = {ξ : E → R : ξ is linear, continuous and inf ξ(Q) < sup ξ(Q)}.

If IntQ ≠ ∅ then Q^{⋆} = E^{∗} \ {0} (where E^{∗} denotes the dual of E). A point q_{o} ∈ Q is a support point of Q if there exists ξ ∈ Q^{⋆} such that ξ(q_{o}) = sup ξ(Q) and we say that ξ supports Q at q_{o}. An element of Q which is not a support point is called a quasi-interior point of Q. Note that some authors defined support points of Q considering all elements of E^{⋆} \ {0} instead of Q^{⋆}. In this case the set of support points can be bigger, but the notion is less subtle. The notion of quasi-interior pionts is a nice tool in the investigation of the properties of the positive cones and positive part of the ball of Banach Lattices or spaces with similar structure (i. e. the set of positive operators acting on Hilbert space).