ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 46, 49 Kruger, Alexander Ya. , International Institute of Labour and Social Relations, Minsk, Belarus On extremality of set systems Let X be a Banach space and let Ω_{1}, Ω_{2} be nonempty closed sets in X. Let us introduce for them a constant θ(Ω_{1}, Ω_{2}) = sup{r > 0 : B_{r} ⊂ Ω_{1} − Ω_{2}}, describing the measure of their “overlapping”. Let us say that the sets Ω_{1}, Ω_{2} form an extremal system in a neighborhood of a point x ∈ Ω_{1} ∩ Ω_{2} if

lim inf_{{ω_{1}∈Ω_{1}, ω_{2}∈Ω_{2} \atop ω_{1}→x, ω_{2}→x,} ; ρ→+0} \frac{θ([Ω_{1} − ω_{1}] ∩ B_{ρ}, [Ω_{2} − ω_{2}] ∩ B_{ρ})}{ρ} = 0.

The necessary and sufficient extremality conditions (in the setting of Asplund spaces) - the generalized Euler equation - are formulated in terms of Fréchet normal cones. They can be viewed as a certain generalization of the classical separation theorem for convex sets.