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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

References

Atsuji, M., ‘Uniform continuity of continuous functions of metric spaces’, Pacific J. Math. 8 (1958), 11–16.CrossRefGoogle Scholar
Beer, G., ‘Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance’, Proc. Amer. Math. Soc. 95 (1985), 653–658.CrossRefGoogle Scholar
Beer, G., ‘More about metric spaces on which continuous functions are uniformly continuous’, Bull. Aust. Math. Soc. 33 (1986), 397–406.CrossRefGoogle Scholar
Beer, G., Bornologies and Lipschitz Analysis (CRC Press, Boca Raton, FL, 2023).CrossRefGoogle Scholar
Beer, G. and Ok, E. A., ‘Embedding of topological posets in hyperspaces’, J. Convex Anal. 30 (2023), 515–540.Google Scholar
Beer, G. and Rice, M., ‘McShane’s condition for the modulus of continuity’, mimeo, 2023.Google Scholar
Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis (AMS Colloquium Publications, RI, 2000).Google Scholar
Bosi, G., Caterino, A. and Ceppitelli, R., ‘Normally preordered spaces and continuous multi-utilities’, Appl. Gen. Topol. 17 (2016), 71–81.CrossRefGoogle Scholar
Calabuig, J., Falciani, H. and Sánchez-Pérez, E., ‘Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial markets’, Neurocomputing 398 (2020), 172–184.CrossRefGoogle Scholar
Carruth, J., ‘A note on partially ordered compacta’, Pacific J. Math. 24 (1968), 229–231.CrossRefGoogle Scholar
Cobzaş, S., Miculescu, R. and Nicolae, A., Lipschitz Functions, Lecture Notes in Mathematics, 2241 (Springer, Cham, 2019).CrossRefGoogle Scholar
Czipser, J. and Geher, L., ‘Extension of functions satisfying a Lipschitz condition’, Acta Math. Acad. Sci. Hungar. 6 (1955), 213–220.CrossRefGoogle Scholar
Dong, M., Yang, X., Wu, Y. and Xue, J.-H., ‘Metric learning via maximizing the Lipschitz margin ratio’, CoRR, 2018.Google Scholar
Evren, O. and Husseinov, F., ‘Extension of monotonic functions and representation of preferences’, Math. Oper. Res. 46 (2021), 1430–1451.CrossRefGoogle Scholar
Evren, O. and Ok, E. A., ‘On the multi-utility representation of preference relations’, J. Math. Econom. 47 (2011), 554–563.CrossRefGoogle Scholar
Falciani, H. and Sánchez-Pérez, E., ‘Semi-Lipschitz functions and machine learning for discrete dynamical systems on graphs’, Mach. Learn. 111 (2022), 1765–1797.CrossRefGoogle Scholar
Fugate, J., Gordh, G. Jr and Lum, L., ‘Arc-smooth continua’, Trans. Amer. Math. Soc. 265 (1981), 545–561.CrossRefGoogle Scholar
Garrido, I. and Jaramillo, J., ‘Lipschitz-type functions on metric spaces’, J. Math. Anal. Appl. 340 (2008), 282–290.CrossRefGoogle Scholar
Gutev, V., ‘Lipschitz extensions and approximations’, J. Math. Anal. Appl. 491 (2020), 1–13.CrossRefGoogle Scholar
Jain, T. and Kundu, S., ‘Atsuji completions: equivalent characterizations’, J. Math. Anal. Appl. 154 (2007), 28–38.Google Scholar
Levy, R. and Rice, M., ‘The approximation and extension of uniformly continuous Banach space valued mappings’, Comment. Math. Univ. Carolin. 2 (1983), 251–265.Google Scholar
McShane, E., ‘Extension of range of functions’, Bull. Amer. Math. Soc. (N.S.) 40 (1934), 837–842.CrossRefGoogle Scholar
Mehta, G., ‘Topological ordered spaces and utility functions’, Internat. Econom. Rev. 18 (1977), 779–782.CrossRefGoogle Scholar
Miculescu, R., ‘Some observations on generalized Lipschitz functions’, Rocky Mountain J. Math. 37 (2007), 893–903.CrossRefGoogle Scholar
Minguzzi, E., ‘Normally preordered spaces and utilities’, Order 30 (2013), 137–150.CrossRefGoogle Scholar
Monteiro, A. and Peixoto, M., ‘Le nombre de Lebesgue et la continuité uniforme’, Port. Math. 10 (1951), 105–113.Google Scholar
Nachbin, L., Topology and Order (van Nostrand, Princeton, NJ, 1965).Google Scholar
Nagata, J., ‘On the uniform topology of bicompactifications’, J. Inst. Polytech Osaka City Univ. 1 (1950), 28–38.Google Scholar
Ok, E. A. and Weaver, N., ‘Lipschitz Bernoulli utility functions’, Math. Oper. Res. 48 (2023), 728–747.CrossRefGoogle Scholar
Ward, L. Jr, ‘A note on Whitney maps’, Canad. Math. Bull. 23 (1980), 373–374.CrossRefGoogle Scholar
Weaver, N., Lipschitz Algebras (World Scientific, NJ, 2018).CrossRefGoogle Scholar
Whitney, H., ‘Analytic extensions of differentiable functions defined in closed sets’, Trans. Amer. Math. Soc. 36 (1934), 63–89.CrossRefGoogle Scholar
Yamazaki, K., ‘Extensions of continuous increasing functions’, Topology Appl. 335 (2023), 108566.CrossRefGoogle Scholar