Uniform Lipschitz-connectedness and metric convexity | ||
| Categories and General Algebraic Structures with Applications | ||
| دوره 22، شماره 1، فروردین 2025، صفحه 139-155 اصل مقاله (492.94 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.48308/cgasa.2024.235538.1489 | ||
| نویسندگان | ||
| Paranjothi Pillay* 1؛ Dharmanand Baboolal2 | ||
| 1Department of Mathematics and Applied Mathematics University of the Western Cape South Africa | ||
| 2School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Metric Space؛ Lipschitz-connected؛ uniformly Lipschitz connected؛ proper metric space؛ dilatations؛ universal Lipschitz constant | ||
| مراجع | ||
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[1] Baboolal, D. and Pillay, P., On uniform Lipschitz-connectedness in metric spaces, Appl. Cat. Str. 17(5) (2009), 487-500. [2] Borwein, J.M., Completeness and the contraction principle, Proc. Amer. Math. Soc. 87(2) (1983), 246-250. [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. | ||
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