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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the axiomatization of convex subsets of Banach spaces


Authors: Valerio Capraro and Tobias Fritz
Journal: Proc. Amer. Math. Soc. 141 (2013), 2127-2135
MSC (2010): Primary 52A01; Secondary 46L10
Published electronically: January 2, 2013
MathSciNet review: 3034438
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Abstract: We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces.


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Valerio Capraro
Affiliation: Institut de Mathématiques, University of Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
Email: valerio.capraro@unine.ch

Tobias Fritz
Affiliation: Institut de Ciències Fotòniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
Email: tobias.fritz@icfo.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11465-6
PII: S 0002-9939(2013)11465-6
Keywords: Convex-like structure, Stone’s barycentric calculus, convex space
Received by editor(s): May 27, 2011
Received by editor(s) in revised form: September 26, 2011
Published electronically: January 2, 2013
Additional Notes: The first author was supported by Swiss SNF Sinergia project CRSI22-130435
The second author was supported by the EU STREP QCS
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.