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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Beyond traditional Curvature-Dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension

Author: Emanuel Milman
Journal: Trans. Amer. Math. Soc. 369 (2017), 3605-3637
MSC (2010): Primary 32F32, 53C21, 39B62, 58J50
Published electronically: December 27, 2016
MathSciNet review: 3605981
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Abstract: We study the isoperimetric, functional and concentration properties of $ n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $ N$ is negative and, more generally, is in the range $ N \in (-\infty ,1)$, extending the scope from the traditional range $ N \in [n,\infty ]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound and discover a new case yielding a single model space (besides the previously known $ N$-sphere and Gaussian measure when $ N \in [n,\infty ]$): a (positively curved) sphere of (possibly negative) dimension $ N \in (-\infty ,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $ N$-dimensional Cheeger isoperimetric inequality and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincaré inequality uniformly for all $ N \in (-\infty ,1-\varepsilon ]$ and enjoy a two-level concentration of the type $ \exp (-\min (t,t^2))$. Our main technical tool is a generalized version of the Heintze-Karcher theorem, which we extend to the range $ N \in (-\infty ,1)$.

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Emanuel Milman
Affiliation: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel

Received by editor(s): March 24, 2015
Received by editor(s) in revised form: July 7, 2015
Published electronically: December 27, 2016
Additional Notes: The author was supported by ISF (grant no. 900/10), BSF (grant no. 2010288) and Marie-Curie Actions (grant no. PCIG10-GA-2011-304066)
Article copyright: © Copyright 2016 by the author

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