{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T08:46:35Z","timestamp":1778575595495,"version":"3.51.4"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T00:00:00Z","timestamp":1757980800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A Takahashi convex structure is considered on Menger PM-spaces and used to investigate the existence of best proximity points for weak cyclic Kannan contractions. We then introduce a concept of a probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger PM-spaces and prove that every compact and convex pair in metrically convex Menger PM-spaces has the probabilistic proximal quasi-normal structure. By applying this geometric property, we survey the existence of a best proximity point for cyclic relatively Kannan nonexpansive maps which preserves distance. In order to provide more accurate results, we obtain the same conclusions in the framework of CAT(0) spaces.<\/jats:p>","DOI":"10.3390\/sym17091549","type":"journal-article","created":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T10:06:25Z","timestamp":1758017185000},"page":"1549","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Best Proximity Theory in Metrically Convex Menger PM-Spaces via Cyclic Kannan Maps"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5439-1631","authenticated-orcid":false,"given":"Moosa","family":"Gabeleh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd 69199-69737, Iran"},{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elif","family":"Uyan\u0131k Ekici","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9640-7846","authenticated-orcid":false,"given":"Maggie","family":"Aphane","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.4064\/sm-2-1-171-180","article-title":"Der Fixpunktsatz in Funktionalriiumen","volume":"2","author":"Schauder","year":"1930","journal-title":"Stud. 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