{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:05:17Z","timestamp":1759334717605,"version":"build-2065373602"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T00:00:00Z","timestamp":1756425600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T00:00:00Z","timestamp":1756425600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"publisher","award":["321896"],"award-info":[{"award-number":["321896"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s00454-025-00757-y","type":"journal-article","created":{"date-parts":[[2025,8,29]],"date-time":"2025-08-29T18:31:33Z","timestamp":1756492293000},"page":"619-660","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fat Triangles Inscribed in Arbitrary Planar Domains"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9592-5150","authenticated-orcid":false,"given":"Ivan Yuri","family":"Violo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,29]]},"reference":[{"key":"757_CR1","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/PL00009337","volume":"19","author":"PK Agarwal","year":"1998","unstructured":"Agarwal, P.K., Amenta, N., Sharir, M.: Largest placement of one convex polygon inside another. Discrete Comput. Geom. 19, 95\u2013104 (1998)","journal-title":"Discrete Comput. Geom."},{"key":"757_CR2","doi-asserted-by":"publisher","first-page":"e7,9","DOI":"10.1017\/fms.2018.7","volume":"6","author":"A Akopyan","year":"2018","unstructured":"Akopyan, A., Avvakumov, S.: Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. Forum Math. Sigma 6, e7,9 (2018)","journal-title":"Forum Math. Sigma"},{"key":"757_CR3","doi-asserted-by":"publisher","first-page":"e5, 16","DOI":"10.1017\/fms.2019.51","volume":"8","author":"J Aslam","year":"2020","unstructured":"Aslam, J., Chen, S., Frick, F., Saloff-Coste, S., Setiabrata, L., Thomas, H.: Splitting loops and necklaces: variants of the square peg problem. Forum Math. Sigma 8, e5, 16 (2020)","journal-title":"Forum Math. Sigma"},{"key":"757_CR4","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF02187904","volume":"3","author":"BS Baker","year":"1988","unstructured":"Baker, B.S., Grosse, E., Rafferty, C.S.: Nonobtuse triangulation of polygons. Discrete Comput. Geom. 3, 147\u2013168 (1988)","journal-title":"Discrete Comput. Geom."},{"key":"757_CR5","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/BF02986206","volume":"29","author":"B Baritompa","year":"2007","unstructured":"Baritompa, B., L\u00f6wen, R., Polster, B., Ross, M.: Mathematical table-turning revisited. Math. Intell. 29, 49\u201358 (2007)","journal-title":"Math. Intell."},{"key":"757_CR6","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1080\/0025570X.1964.11975529","volume":"37","author":"A Bruckner","year":"1964","unstructured":"Bruckner, A.: A note on convex polygons inscribed in open sets. Math. Mag. 37, 250\u2013251 (1964)","journal-title":"Math. Mag."},{"key":"757_CR7","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1112\/blms\/2.1.73","volume":"2","author":"R Fenn","year":"1970","unstructured":"Fenn, R.: The table theorem. Bull. Lond. Math. Soc. 2, 73\u201376 (1970)","journal-title":"Bull. Lond. Math. Soc."},{"issue":"194","key":"757_CR8","first-page":"509","volume":"2","author":"JE Greene","year":"2021","unstructured":"Greene, J.E., Lobb, A.: The rectangular peg problem. Ann. Math. 2(194), 509\u2013517 (2021)","journal-title":"Ann. Math."},{"key":"757_CR9","doi-asserted-by":"crossref","unstructured":"Hebbert, C.M.: The inscribed and circumscribed squares of a quadrilateral and their significance in kinematic geometry. Ann. Math. 2(16), 38\u201342 (1914\/15)","DOI":"10.2307\/1968039"},{"key":"757_CR10","first-page":"22","volume":"69","author":"CO Horgan","year":"2022","unstructured":"Horgan, C.O., Murphy, J.G.: On an angle with magical properties. Not. Am. Math. Soc. 69, 22\u201325 (2022)","journal-title":"Not. Am. Math. Soc."},{"key":"757_CR11","unstructured":"Hugelmeyer, C.: Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to $$\\sqrt{3}$$. arXiv:1803.07417 (2018)"},{"issue":"194","key":"757_CR12","first-page":"497","volume":"2","author":"C Hugelmeyer","year":"2021","unstructured":"Hugelmeyer, C.: Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios. Ann. Math. 2(194), 497\u2013508 (2021)","journal-title":"Ann. Math."},{"key":"757_CR13","doi-asserted-by":"publisher","first-page":"45","DOI":"10.4171\/em\/442","volume":"76","author":"D Ismailescu","year":"2021","unstructured":"Ismailescu, D., Kim, M.J., Wang, E.: On the maximum area of inscribed polygons. Elem. Math. 76, 45\u201361 (2021)","journal-title":"Elem. Math."},{"key":"757_CR14","doi-asserted-by":"publisher","first-page":"967","DOI":"10.4169\/amer.math.monthly.122.10.967","volume":"122","author":"J Jer\u00f3nimo\u00a0Castro","year":"2015","unstructured":"Jer\u00f3nimo\u00a0Castro, J.: On convex curves which have many inscribed triangles of maximum area. Am. Math. Mon. 122, 967\u2013971 (2015)","journal-title":"Am. Math. Mon."},{"issue":"24","key":"757_CR15","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1112\/jlms\/s2-24.1.182","volume":"2","author":"EH Kronheimer","year":"1981","unstructured":"Kronheimer, E.H., Kronheimer, P.B.: The tripos problem. J. Lond. Math. Soc. 2(24), 182\u2013192 (1981)","journal-title":"J. Lond. Math. Soc."},{"key":"757_CR16","doi-asserted-by":"publisher","first-page":"133","DOI":"10.5486\/PMD.2004.2953","volume":"65","author":"M Lassak","year":"2004","unstructured":"Lassak, M.: On relatively equilateral polygons inscribed in a convex body. Publ. Math. Debrecen 65, 133\u2013148 (2004)","journal-title":"Publ. Math. Debrecen"},{"key":"757_CR17","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s13366-011-0026-x","volume":"52","author":"M Lassak","year":"2011","unstructured":"Lassak, M.: Approximation of convex bodies by inscribed simplices of maximum volume. Beitr. Algebra Geom. 52, 389\u2013394 (2011)","journal-title":"Beitr. Algebra Geom."},{"key":"757_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2021.101792","volume":"98","author":"S Lee","year":"2021","unstructured":"Lee, S., Eom, T., Ahn, H.-K.: Largest triangles in a polygon. Comput. Geom. 98, 101792 (2021)","journal-title":"Comput. Geom."},{"key":"757_CR19","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1006\/eujc.2001.0531","volume":"23","author":"H Maehara","year":"2002","unstructured":"Maehara, H.: Acute triangulations of polygons. Eur. J. Combin. 23, 45\u201355 (2002)","journal-title":"Eur. J. Combin."},{"key":"757_CR20","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1137\/S009753979018330X","volume":"23","author":"J Matou\u0161ek","year":"1994","unstructured":"Matou\u0161ek, J., Pach, J., Sharir, M., Sifrony, S., Welzl, E.: Fat triangles determine linearly many holes. SIAM J. Comput. 23, 154\u2013169 (1994)","journal-title":"SIAM J. Comput."},{"key":"757_CR21","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1090\/noti1100","volume":"61","author":"B Matschke","year":"2014","unstructured":"Matschke, B.: A survey on the square peg problem. Not. Am. Math. Soc. 61, 346\u2013352 (2014)","journal-title":"Not. Am. Math. Soc."},{"key":"757_CR22","doi-asserted-by":"publisher","first-page":"5719","DOI":"10.1090\/tran\/8359","volume":"374","author":"B Matschke","year":"2021","unstructured":"Matschke, B.: Quadrilaterals inscribed in convex curves. Trans. Am. Math. Soc. 374, 5719\u20135738 (2021)","journal-title":"Trans. Am. Math. Soc."},{"key":"757_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4064\/fm-110-1-1-9","volume":"110","author":"MD Meyerson","year":"1980","unstructured":"Meyerson, M.D.: Equilateral triangles and continuous curves. Fund. Math. 110, 1\u20139 (1980)","journal-title":"Fund. Math."},{"key":"757_CR24","doi-asserted-by":"publisher","first-page":"167","DOI":"10.2140\/pjm.1981.97.167","volume":"97","author":"MD Meyerson","year":"1981","unstructured":"Meyerson, M.D.: Convexity and the table theorem. Pac. J. Math. 97, 167\u2013169 (1981)","journal-title":"Pac. J. Math."},{"key":"757_CR25","doi-asserted-by":"publisher","first-page":"167","DOI":"10.2140\/pjm.1984.110.167","volume":"110","author":"MD Meyerson","year":"1984","unstructured":"Meyerson, M.D.: Remarks on Fenn\u2019s \u201cthe table theorem\u2019\u2019 and Zaks\u2019 \u201cthe chair theorem\u2019\u2019. Pac. J. Math. 110, 167\u2013169 (1984)","journal-title":"Pac. J. Math."},{"key":"757_CR26","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/BF00151519","volume":"43","author":"MJ Nielsen","year":"1992","unstructured":"Nielsen, M.J.: Triangles inscribed in simple closed curves. Geom. Dedicata 43, 291\u2013297 (1992)","journal-title":"Geom. Dedicata"},{"key":"757_CR27","doi-asserted-by":"publisher","first-page":"722","DOI":"10.1007\/s00454-014-9578-5","volume":"51","author":"VH Pettersson","year":"2014","unstructured":"Pettersson, V.H., Tverberg, H.A., \u00d6sterg\u00e5rd, P.R.J.: A note on Toeplitz\u2019 conjecture. Discrete Comput. Geom. 51, 722\u2013728 (2014)","journal-title":"Discrete Comput. Geom."},{"key":"757_CR28","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10711-020-00516-8","volume":"208","author":"RE Schwartz","year":"2020","unstructured":"Schwartz, R.E.: A trichotomy for rectangles inscribed in Jordan loops. Geom. Dedicata 208, 177\u2013196 (2020)","journal-title":"Geom. Dedicata"},{"key":"757_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/bull\/1755","volume":"59","author":"RE Schwartz","year":"2022","unstructured":"Schwartz, R.E.: Rectangles, curves, and Klein bottles. Bull. Am. Math. Soc. (N.S.) 59, 1\u201317 (2022)","journal-title":"Bull. Am. Math. Soc. (N.S.)"},{"key":"757_CR30","doi-asserted-by":"publisher","first-page":"e30, 63","DOI":"10.1017\/fms.2017.23","volume":"5","author":"T Tao","year":"2017","unstructured":"Tao, T.: An integration approach to the Toeplitz square peg problem. Forum Math. Sigma 5, e30, 63 (2017)","journal-title":"Forum Math. Sigma"},{"key":"757_CR31","unstructured":"Toeplitz, O.: Equilateral triangles and continuous curves. \u00dcber einige Aufgaben der Analysis situs 4 (1911)"},{"key":"757_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2020.105943","volume":"161","author":"I van der Hoog","year":"2020","unstructured":"van der Hoog, I., Keikha, V., L\u00f6ffler, M., Mohades, A., Urhausen, J.: Maximum-area triangle in a convex polygon, revisited. Inf. Process. Lett. 161, 105943 (2020)","journal-title":"Inf. Process. Lett."},{"key":"757_CR33","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1007\/s00454-005-1188-9","volume":"34","author":"L Yuan","year":"2005","unstructured":"Yuan, L.: Acute triangulations of polygons. Discrete Comput. Geom. 34, 697\u2013706 (2005)","journal-title":"Discrete Comput. Geom."}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00757-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-025-00757-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00757-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T20:03:29Z","timestamp":1759262609000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-025-00757-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,29]]},"references-count":33,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["757"],"URL":"https:\/\/doi.org\/10.1007\/s00454-025-00757-y","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2025,8,29]]},"assertion":[{"value":"15 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}