
Journal of Convex Analysis 28 (2021), No. 1, 031040 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 RadonNikodym 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 Avalued quasimartingales with sharply divergent behavior. Keywords: Dentable sets in normed spaces, martingale convergence, RadonNikodym property, convex sets, extreme points. MSC: 46B22; 46B20, 52A07, 60G42. [ Fulltextpdf (123 KB)] for subscribers only. 