{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:03:28Z","timestamp":1760148208479,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universitat Polit\u00e8cnica de Val\u00e8ncia","award":["PAID-01-21","PID2020-112759GB-I00"],"award-info":[{"award-number":["PAID-01-21","PID2020-112759GB-I00"]}]},{"name":"MCIN\/AEI\/10.13039\/501100011033","award":["PAID-01-21","PID2020-112759GB-I00"],"award-info":[{"award-number":["PAID-01-21","PID2020-112759GB-I00"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p\u2013average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p=2, explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given.<\/jats:p>","DOI":"10.3390\/axioms12040359","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T03:24:18Z","timestamp":1681097058000},"page":"359","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression"],"prefix":"10.3390","volume":"12","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\/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":[[2023,4,7]]},"reference":[{"key":"ref_1","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. 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