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

 

Does negative type characterize the round sphere?


Author: Simon Lyngby Kokkendorff
Journal: Proc. Amer. Math. Soc. 135 (2007), 3695-3702
MSC (2000): Primary 51K99, 53C35, 31C99
Published electronically: August 7, 2007
MathSciNet review: 2336586
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Abstract: We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio $ 1$ must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio $ 1$ must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.


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Additional Information

Simon Lyngby Kokkendorff
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Kgs. Lyngby, Denmark
Email: S.L.Kokkendorff@mat.dtu.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08951-4
PII: S 0002-9939(07)08951-4
Received by editor(s): August 24, 2006
Published electronically: August 7, 2007
Additional Notes: The author was supported by the Danish Research Agency
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.