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Proceedings of the American Mathematical Society

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

 

 

Injectivity of quasi-isometric mappings of balls


Author: Julian Gevirtz
Journal: Proc. Amer. Math. Soc. 85 (1982), 345-349
MSC: Primary 47H99; Secondary 46B99
DOI: https://doi.org/10.1090/S0002-9939-1982-0656099-X
MathSciNet review: 656099
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Abstract: Let $ X$ and $ Y$ be real Banach spaces. A mapping $ f$ of an open subset $ R$ of $ X$ into $ Y$ is said to be $ (m,M)$-isometric if it is a local homeomorphism for which $ M \geqslant {D^ + }f(x)$ and $ m \leqslant {D^ - }f(x)$ for all $ x$ in $ R$, where $ {D^ + }f(x)$ and $ {D^ - }f(x)$ are, respectively, the upper and lower limits of $ \vert f(y) - f(x)\vert/\vert y - x\vert$ as $ y \to x$. We show that if $ R$ is a ball then all $ (m,M)$isometric mappings of $ R$ are injective provided that $ M/m < 1.114 \ldots $ and we also give some numerical improvements of similar results of F. John for the special case that $ X$ is a Hilbert space.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0656099-X
Keywords: Quasi-isometric mapping, injectivity
Article copyright: © Copyright 1982 American Mathematical Society