{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T16:22:07Z","timestamp":1777479727929,"version":"3.51.4"},"reference-count":16,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,8,1]],"date-time":"2024-08-01T00:00:00Z","timestamp":1722470400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,8,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article we present the Mizar proof of the isoperimetric theorem (one of the theorems listed among Wiedijk\u2019s Top 100 mathematical theorems), inspired by Peter D. Lax\u2019s paper \u201cA Short Path to the Shortest Path\u201d. Using relatively simple formal apparatus of continuous and differentiable functions, we show that among all curves of fixed length connecting two points on the x-axis, a semicircle is the curve which maximizes the area between the curve and the x-axis.<\/jats:p>","DOI":"10.2478\/forma-2024-0015","type":"journal-article","created":{"date-parts":[[2025,1,3]],"date-time":"2025-01-03T14:07:48Z","timestamp":1735913268000},"page":"187-194","source":"Crossref","is-referenced-by-count":1,"title":["Classical Isoperimetric Theorem"],"prefix":"10.2478","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1110-4342","authenticated-orcid":false,"given":"Kazuhisa","family":"Nakasho","sequence":"first","affiliation":[{"name":"Yamaguchi University , Yamaguchi , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Karuizawa Hotch 244-1, Nagano , Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"key":"2026042801415761426_j_forma-2024-0015_ref_001","doi-asserted-by":"crossref","unstructured":"Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026042801415761426_j_forma-2024-0015_ref_002","doi-asserted-by":"crossref","unstructured":"Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801415761426_j_forma-2024-0015_ref_003","doi-asserted-by":"crossref","unstructured":"Viktor Bl\u00e5sj\u00f6. The isoperimetric problem. The American Mathematical Monthly, 112(6): 526\u2013566, 2005.","DOI":"10.1080\/00029890.2005.11920227"},{"key":"2026042801415761426_j_forma-2024-0015_ref_004","doi-asserted-by":"crossref","unstructured":"Noboru Endou. Differentiation on interval. Formalized Mathematics, 31(1):9\u201321, 2023. doi:10.2478\/forma-2023-0002.","DOI":"10.2478\/forma-2023-0002"},{"key":"2026042801415761426_j_forma-2024-0015_ref_005","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Multidimensional measure space and integration. Formalized Mathematics, 31(1):181\u2013192, 2023. doi:10.2478\/forma-2023-0017.","DOI":"10.2478\/forma-2023-0017"},{"key":"2026042801415761426_j_forma-2024-0015_ref_006","doi-asserted-by":"crossref","unstructured":"Noboru Endou and Yasunari Shidama. Integral of continuous functions of two variables. 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The HOL Light system reference. 2023. http:\/\/www.cl.cam.ac.uk\/~jrh13\/hol-light\/reference.pdf."},{"key":"2026042801415761426_j_forma-2024-0015_ref_010","unstructured":"John Harrison. The isoperimetric inequality. 2023. Available online at https:\/\/github.com\/jrh13\/hol-light\/blob\/master\/100\/isoperimetric.ml."},{"key":"2026042801415761426_j_forma-2024-0015_ref_011","unstructured":"Andreas Hehl. The isoperimetric inequality. Proseminar Curves and Surfaces, Universitaet Tuebingen, Tuebingen, 2013."},{"key":"2026042801415761426_j_forma-2024-0015_ref_012","doi-asserted-by":"crossref","unstructured":"Peter David Lax. A short path to the shortest path. The American Mathematical Monthly, 102(2):158\u2013159, 1995.","DOI":"10.1080\/00029890.1995.11990550"},{"key":"2026042801415761426_j_forma-2024-0015_ref_013","doi-asserted-by":"crossref","unstructured":"Robert Osserman. The isoperimetric inequality. 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