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

 

Relations between geometric convexity, doubling measures and property $ \Gamma$


Authors: Luis A. Caffarelli and Michael G. Crandall
Journal: Proc. Amer. Math. Soc. 142 (2014), 2395-2406
MSC (2010): Primary 46B20
Published electronically: March 21, 2014
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Abstract: In this article it is shown that the three conditions on the norm $ \left \Vert\cdot \right \Vert$ of a Banach space called ``geometric convexity'', ``balanced'' and ``doubling'' in an earlier work by the authors related to eikonal equations are in fact all equivalent. Moreover, each of them is equivalent to a condition called ``Property $ \Gamma $'' by Ganichev and Kalton. A fifth condition, that the second derivative of the function $ t\mapsto \left \Vert x+ty\right \Vert$ is a doubling measure on $ [-2,2]$ for suitable $ x, y\in X,$ is also equivalent to the various other properties, and this formulation occupies a central place in the analysis.


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Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
Email: caffarel@math.utexas.eduu

Michael G. Crandall
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106
Email: crandall@math.ucsb.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11940-X
PII: S 0002-9939(2014)11940-X
Keywords: Banach space, Property $\Gamma$, geometric convexity, doubling measures
Received by editor(s): September 9, 2011
Received by editor(s) in revised form: June 25, 2012, and July 11, 2012
Published electronically: March 21, 2014
Additional Notes: The first author was supported in part by NSF Grant DMS-1160802.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society