Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-2014-11940-X
Published electronically: March 21, 2014
MathSciNet review: 3195762
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article it is shown that the three conditions on the norm $\left \|\cdot \right \|$ 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 $t\mapsto \left \|x+ty\right \|$ 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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46B20

Retrieve articles in all journals with MSC (2010): 46B20


Additional Information

Luis A. Caffarelli
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
MR Author ID: 44175
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

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