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

 

The relationships of spans of convex continua in $ {\bf R}\sp n$


Author: Thelma West
Journal: Proc. Amer. Math. Soc. 111 (1991), 261-265
MSC: Primary 54F15
MathSciNet review: 1037227
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Abstract: It has been conjectured that $ {\sigma ^ * }(X) \geq \tfrac{1}{2}\sigma _0^ * (X)$ for each nonempty connected metric space $ X$. In this paper we show that $ {\sigma ^ * }(X) \geq \tfrac{{\sqrt {13} }}{4}\sigma _0^ * (X)$ for each convex continuum $ X$ in $ {R^n}$. We also show that under certain conditions a lower bound for the ratio $ {\sigma ^ * }(X)/\sigma _0^ * (X)$ is larger than $ \tfrac{{\sqrt {13} }}{4}$. It has also been conjectured that $ {\sigma ^ * }(X) \geq \sigma (X)/2$ and that $ \sigma _0^ * (X) \geq {\sigma _0}(X)/2$ for each nonempty connected metric space $ X$. We show that these two inequalities hold when $ X$ is a convex continuum in $ {R^n}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1037227-9
PII: S 0002-9939(1991)1037227-9
Keywords: Span, convex continua
Article copyright: © Copyright 1991 American Mathematical Society