Antiproximinal sets in spaces of continuous functions

Closed convex bounded antiproximinal bodies are constructed in the infinite-dimensional spacesC(Q), C₀(T), L(S, S, ), andB(S), whereQ is a topological space andT is a locally compact Hausdorff space. It is shown that there are no closed bounded antiproximinal sets in Banach spaces with the Radon-Nikodym property.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 643–657, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00196.