Uniform Lipschitz-connectedness and metric convexity

Document Type : Research Paper

Authors

1 Department of Mathematics and Applied Mathematics University of the Western Cape South Africa

2 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.

10.48308/cgasa.2024.235538.1489

Abstract

In this paper we continue with our study of uniformly Lipschitz-connected metric spaces.   We obtain further properties of uniformly Lipschitz-connected metric spaces and then obtain a generalisation of a result due to Edelstein.  In addition, we show that for a proper Lipschitz-connected metric space,  $L_d = 1$ precisely when $X$ is convex, which leads us to conjecture that $L_d$ is a kind of measure of convexity in a proper Lipschitz-connected metric space.  We provide some examples to corroborate our conjecture.

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References

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[3] Burago, D., Burago, Y., and Ivanov, S., “A Course in Metric Geometry”, American Mathematical Society, 2001.
[4] Edelstein, M., An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12(1) (1961), 7-10.
[5] Menger, K., Untersuchungen ¨uber allgemeine metrik, Math. Ann. 100 (1928), 75-163.
[6] Papadopoulos, A., “Metric Spaces, Convexity and Nonpositive Curvature”, IRMA Lectures in Mathematics and Theoretical Physics 6, 2005.
Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 03 September 2024

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