Geodesic
Nondentable Sets in Banach Spaces
Journal of convex analysis, Tome 28 (2021) no. 1, pp. 31-4
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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.
Classification : 46B22, 46B20, 52A07, 60G42
Mots-clés : Dentable sets in normed spaces, martingale convergence, Radon-Nikodym property, convex sets, extreme points 
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     author = {S. J. Dilworth and C. Gartland and D. Kutzarova and N. L. Randrianarivony},
     title = {Nondentable {Sets} in {Banach} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {31--4},
     year = {2021},
     volume = {28},
     number = {1},
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}
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S. J. Dilworth; C. Gartland; D. Kutzarova; N. L. Randrianarivony. Nondentable Sets in Banach Spaces. Journal of convex analysis, Tome 28 (2021) no. 1, pp. 31-4. http://geodesic.mathdoc.fr/item/JCA_2021_28_1_JCA_2021_28_1_a3/