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On Aumann's Theorem that the sphere does not admit a mean

F. Javier Trigos-Arrieta·Marian Turzanski
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Abstract

We prove that the circle S_1 does not have a 2-mean, i.e., S_1 times S_1 cannot have a retraction r onto its diagonal with r(x,y) = r(y,x), whenever x,y in S_1. Our proof is combinatorial and topological rather than analytical.

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F. Javier Trigos-Arrieta, Marian Turzanski. 2005-01-19. On Aumann's Theorem that the sphere does not admit a mean. https://arxiv.org/abs/math/0501305

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