Journal of Convex Analysis 28 (2021), No. 1, 031--040
Copyright Heldermann Verlag 2021
Nondentable Sets in Banach Spaces
Stephen J. Dilworth
Dept. of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
dilworth@math.sc.edu
Chris Gartland
Dept. of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
cgartla2@illinois.edu
Denka Kutzarova
Dept. of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
and: Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
denka@math.uiuc.edu
N. Lovasoa Randrianarivony
Dept. of Mathematics and Statistics, Saint Louis University, St. Louis, MO 63103, U.S.A.
nrandria@slu.edu
In his study of the Radon-Nikodym property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set A that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set A (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of A-valued quasimartingales with sharply divergent behavior.
Keywords: Dentable sets in normed spaces, martingale convergence, Radon-Nikodym property, convex sets, extreme points.
MSC: 46B22; 46B20, 52A07, 60G42.
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