TY - JOUR
T1 - Nondentable Sets in Banach Spaces
AU - Dilworth, Stephen J.
AU - Gartland, Chris
AU - Kutzarova, Denka
AU - Randrianarivony, N. Lovasoa
N1 - Funding Information:
∗The third author was supported by Simons Foundation Collaborative Grant No 636954. †The fourth author was partially supported by NSF grant DMS-1301591.
Publisher Copyright:
© Heldermann Verlag
PY - 2021
Y1 - 2021
N2 - In his study of the Radon-Nikodým 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.
AB - In his study of the Radon-Nikodým 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.
KW - Convex sets
KW - Dentable sets in normed spaces
KW - Extreme points
KW - Martingale convergence
KW - Radon-Nikodým property
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M3 - Article
AN - SCOPUS:85099665838
SN - 0944-6532
VL - 28
SP - 31
EP - 40
JO - Journal of Convex Analysis
JF - Journal of Convex Analysis
IS - 1
ER -