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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Kakutani's theorem for real-valued maps on a compact surface


Authors: E. H. Kronheimer and P. B. Kronheimer
Journal: Proc. Amer. Math. Soc. 85 (1982), 256-260
MSC: Primary 53A05
DOI: https://doi.org/10.1090/S0002-9939-1982-0652453-0
MathSciNet review: 652453
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Abstract: Let $ M$ be a compact $ 2$-manifold (without boundary) $ {C^1}$-embedded in $ {{\mathbf{R}}^3}$. Then there exists positive $ \sigma $ such that, given any positive $ \tau \leqslant \sigma $ and any continuous map $ f:M \to {\mathbf{R}}$, there exist points $ p$,$ q$,$ r \in M$, satisfying $ \left\Vert {q - r} \right\Vert = \left\Vert {r - p} \right\Vert = \left\Vert {p - q} \right\Vert = \tau $ in the euclidean norm, for which $ f(p) = f(q) = f(r)$.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0652453-0
Article copyright: © Copyright 1982 American Mathematical Society

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