{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:08:14Z","timestamp":1760058494824,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,6]],"date-time":"2025-04-06T00:00:00Z","timestamp":1743897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071444","12201581","202403021221109","20210302124657","202103021224291","202303021221116"],"award-info":[{"award-number":["12071444","12201581","202403021221109","20210302124657","202103021224291","202303021221116"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Research Program of Shanxi Province","award":["12071444","12201581","202403021221109","20210302124657","202103021224291","202303021221116"],"award-info":[{"award-number":["12071444","12201581","202403021221109","20210302124657","202103021224291","202303021221116"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For each n-dimensional real Banach space X and each positive integer m, let \u03b2(X,m) be the infimum of \u03b4\u2208(0,1] such that each set A\u2286X having diameter 1 can be represented as the union of m subsets of A, whose diameters are not greater than \u03b4. Providing accurate estimations of \u03b2(X,m) for specific choices of X and m is crucial for addressing the extension of the classical Borsuk\u2019s problem. A general framework for estimating \u03b2(X,m) via constructing and refining universal covering systems is presented. As an example, a universal covering system is constructed in \u211313 and it is shown that \u03b2(\u211313,8)\u226411\/12 by a feasible partitioning of members in this universal covering system.<\/jats:p>","DOI":"10.3390\/axioms14040277","type":"journal-article","created":{"date-parts":[[2025,4,7]],"date-time":"2025-04-07T05:30:59Z","timestamp":1744003859000},"page":"277","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Universal Covering System and Borsuk\u2019s Problem in Finite Dimensional Banach Spaces"],"prefix":"10.3390","volume":"14","author":[{"given":"Xincong","family":"Qi","sequence":"first","affiliation":[{"name":"School of Mathematics, North University of China, Taiyuan 030051, China"}]},{"given":"Xinling","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Harbin Institute of Technology, Harbin 150001, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-1809-1448","authenticated-orcid":false,"given":"Yunfang","family":"Lyu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2499-2836","authenticated-orcid":false,"given":"Senlin","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics, North University of China, Taiyuan 030051, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s43037-023-00266-1","article-title":"Complete sets in normed linear spaces","volume":"17","author":"He","year":"2023","journal-title":"Banach J. 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