{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:15Z","timestamp":1760144175522,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T00:00:00Z","timestamp":1710374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MCIN\/AEI\/10.13039\/501100011033","award":["PID2020-112759GB-I00","PAID-01-21"],"award-info":[{"award-number":["PID2020-112759GB-I00","PAID-01-21"]}]},{"name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","award":["PID2020-112759GB-I00","PAID-01-21"],"award-info":[{"award-number":["PID2020-112759GB-I00","PAID-01-21"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function \u03d5. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modeling of livability indices in North American cities.<\/jats:p>","DOI":"10.3390\/axioms13030192","type":"journal-article","created":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T05:43:30Z","timestamp":1710395010000},"page":"192","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Moduli of Continuity in Metric Models and Extension of Livability Indices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2544-8875","authenticated-orcid":false,"given":"Roger","family":"Arnau","sequence":"first","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8398-8664","authenticated-orcid":false,"given":"Jose M.","family":"Calabuig","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2328-2173","authenticated-orcid":false,"given":"\u00c1lvaro","family":"Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8854-3154","authenticated-orcid":false,"given":"Enrique A.","family":"S\u00e1nchez P\u00e9rez","sequence":"additional","affiliation":[{"name":"Instituto Universitario de Matem\u00e1tica Pura y Aplicada, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,14]]},"reference":[{"key":"ref_1","first-page":"803","article-title":"Index spaces and standard indices in metric modelling","volume":"27","year":"2022","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.neucom.2020.02.052","article-title":"Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial markets","volume":"398","author":"Calabuig","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"M\u00e9moli, F., Sapiro, G., and Thompson, P.M. (2006, January 9). Geometric surface and brain warping via geodesic minimizing Lipschitz extensions. Proceedings of the 1st MICCAI Workshop on Mathematical Foundations of Computational Anatomy: Geometrical, Statistical and Registration Methods for Modeling Biological Shape Variability, Munich, Germany.","DOI":"10.21236\/ADA478383"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.matpur.2005.04.005","article-title":"Extension theorems for vector valued maps","volume":"85","author":"Dacorogna","year":"2006","journal-title":"J. Math. 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(2019). Lipschitz Functions, Springer International Publishing.","DOI":"10.1007\/978-3-030-16489-8"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1090\/S0002-9904-1934-05978-0","article-title":"Extension of range of functions","volume":"40","author":"McShane","year":"1934","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_11","unstructured":"Whitney, H. (1992). Hassler Whitney Collected Papers, Springer."},{"key":"ref_12","unstructured":"Efimov, A. (2023, October 22). Modulus of Continuity, Encyclopaedia of Mathematics. Available online: https:\/\/encyclopediaofmath.org\/wiki\/Continuity."},{"key":"ref_13","first-page":"57","article-title":"Absolutely minimizing Lipschitz extensions on a metric space","volume":"27","author":"Juutinen","year":"2002","journal-title":"Ann. Acad. Sci. Fenn. Math."},{"key":"ref_14","unstructured":"Ritchie, H., and Roser, M. (2018). Urbanization. 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