Does negative type characterize the round sphere?

Kokkendorff, Simon

doi:10.1090/S0002-9939-07-08951-4

Does negative type characterize the round sphere?
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by Simon Lyngby Kokkendorff PDF

Proc. Amer. Math. Soc. 135 (2007), 3695-3702 Request permission

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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