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ORDER-PRESERVING EXTENSIONS OF LIPSCHITZ MAPS

Part of: Maps and general types of spaces defined by maps Ordered structures

Published online by Cambridge University Press:  17 December 2024

EFE A. OK
EFE A. OK*
Affiliation:
Department of Economics and Courant Institute of Mathematical Studies, New York University, New York, New York, USA
*
e-mail: [email protected]
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Abstract

We study the problem of extending an order-preserving real-valued Lipschitz map defined on a subset of a partially ordered metric space without increasing its Lipschitz constant and preserving its monotonicity. We show that a certain type of relation between the metric and order of the space, which we call radiality, is necessary and sufficient for such an extension to exist. Radiality is automatically satisfied by the equality relation, so the classical McShane–Whitney extension theorem is a special case of our main characterization result. As applications, we obtain a similar generalization of McShane’s uniformly continuous extension theorem, along with some functional representation results for radial partial orders.

Keywords

partially ordered metric spacesorder-preserving Lipschitz mapsradial convexityMcShane–Whitney extension theoremextension of uniformly continuous functions

MSC classification

Primary: 54C20: Extension of maps 54C30: Real-valued functions
Secondary: 06F30: Topological lattices, order topologies
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 118 , Issue 1 , February 2025 , pp. 91 - 107
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by George Willis

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