{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:44:11Z","timestamp":1760402651159,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,4]],"date-time":"2022-01-04T00:00:00Z","timestamp":1641254400000},"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":["11971493\uff0c12071491"],"award-info":[{"award-number":["11971493\uff0c12071491"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we firstly investigate the constant H(X) proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant GL(X) closely related to H(X), along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms GL(X,p) and CL(X) explicitly.<\/jats:p>","DOI":"10.3390\/sym14010072","type":"journal-article","created":{"date-parts":[[2022,1,9]],"date-time":"2022-01-09T23:35:09Z","timestamp":1641771309000},"page":"72","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2641-3773","authenticated-orcid":false,"given":"Bingren","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6049-5282","authenticated-orcid":false,"given":"Qi","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4322-308X","authenticated-orcid":false,"given":"Yongjin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ahmad, A., Liu, Q., and Li, Y. (2021). Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls. Symmetry, 13.","DOI":"10.3390\/sym13071294"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"275","DOI":"10.4064\/sm144-3-5","article-title":"On James Jordan\u2013von Neumann constants and the normal structure coefficient of Banach spaces","volume":"144","author":"Kato","year":"2001","journal-title":"Stud. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s40840-021-01196-7","article-title":"On New Moduli Related to the Generalization of the Parallelogram Law","volume":"45","author":"Liu","year":"2021","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s00605-021-01540-w","article-title":"The von Neumann\u2013Jordan and another constants in Radon planes","volume":"195","author":"Mizuguchi","year":"2021","journal-title":"Monatsh. Math."},{"key":"ref_5","first-page":"341","article-title":"On James type constants and the normal structure in Banach spaces","volume":"23","author":"Zuo","year":"2020","journal-title":"Math. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"542","DOI":"10.2307\/1970663","article-title":"Uniformly non-square Banach spaces","volume":"80","author":"James","year":"1964","journal-title":"Ann. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1090\/S0002-9947-1936-1501880-4","article-title":"Uniformly convex spaces","volume":"40","author":"Clarkson","year":"1936","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1017\/S1446788700035230","article-title":"On the geometry of spheres in normed linear spaces","volume":"48","author":"Gao","year":"1990","journal-title":"J. Aust. Math. Soc. Ser. A."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"114","DOI":"10.2307\/1968512","article-title":"The von Neumann\u2013Jordan constant for the Lebesgue space","volume":"38","author":"Clarkson","year":"1937","journal-title":"Ann. Math."},{"key":"ref_10","first-page":"253","article-title":"An inequality of M. R\u0103dulescu and S. R\u0103dulescu which characterizes inner product spaces","volume":"47","year":"2001","journal-title":"Rev. Roum. Math. Pures Appl."},{"key":"ref_11","first-page":"135","article-title":"A counterexample for a conjecture of G. Zb\u0103ganu about the Neumann\u2013Jordan constant","volume":"51","author":"Alonso","year":"2006","journal-title":"Rev. Roum. Math. Pures Appl."},{"key":"ref_12","first-page":"241","article-title":"Normal hexagon and more general Banach spaces with uniform normal structure","volume":"20","author":"Gao","year":"2000","journal-title":"J. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/72\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:00:07Z","timestamp":1760364007000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/72"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,4]]},"references-count":12,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010072"],"URL":"https:\/\/doi.org\/10.3390\/sym14010072","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,4]]}}}