{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:50:48Z","timestamp":1740124248708,"version":"3.37.3"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T00:00:00Z","timestamp":1675036800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T00:00:00Z","timestamp":1675036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100020618","name":"Universit\u00e4t Bayreuth","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100020618","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A cover for a family <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {F}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>F<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of sets in the plane is a set into which every set in <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {F}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>F<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be isometrically moved. We are interested in the convex cover of smallest area for a given family of triangles. Park and Cheong conjectured that any family of triangles of bounded diameter has a smallest convex cover that is itself a triangle. The conjecture is equivalent to the claim that for every convex set\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {X}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> there is a triangle\u00a0<jats:italic>Z<\/jats:italic> whose area is not larger than the area of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {X}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, such that\u00a0<jats:italic>Z<\/jats:italic> covers the family of triangles contained in\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {X}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove this claim for the case where a diameter of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\mathcal {X}}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> lies on its boundary. We also give a complete characterization of the smallest convex cover for the family of triangles contained in a half-disk, and for the family of triangles contained in a square. In both cases, this cover is a triangle.<\/jats:p>","DOI":"10.1007\/s10998-022-00503-4","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T13:05:23Z","timestamp":1675083923000},"page":"86-109","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Covering families of triangles"],"prefix":"10.1007","volume":"87","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4467-7075","authenticated-orcid":false,"given":"Otfried","family":"Cheong","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4275-5068","authenticated-orcid":false,"given":"Olivier","family":"Devillers","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6914-1651","authenticated-orcid":false,"given":"Marc","family":"Glisse","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2490-1359","authenticated-orcid":false,"given":"Ji-won","family":"Park","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,30]]},"reference":[{"key":"503_CR1","doi-asserted-by":"publisher","first-page":"288","DOI":"10.20382\/jocg.v6i1a12","volume":"6","author":"JC Baez","year":"2015","unstructured":"J.C. Baez, K. Bagdasaryan, P. Gibbs, The Lebesgue universal covering problem. J. Comput. Geom. 6, 288\u2013299 (2015). https:\/\/doi.org\/10.20382\/jocg.v6i1a12","journal-title":"J. Comput. Geom."},{"key":"503_CR2","volume-title":"Research Problems in Discrete Geometry","author":"P Brass","year":"2005","unstructured":"P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, Berlin, 2005)"},{"key":"503_CR3","doi-asserted-by":"publisher","first-page":"537","DOI":"10.1142\/S0218195905001828","volume":"15","author":"P Brass","year":"2005","unstructured":"P. Brass, M. Sharifi, A lower bound for Lebesgue\u2019s universal cover problem. Int. J. Comput. Geom. Appl. 15, 537\u2013544 (2005). https:\/\/doi.org\/10.1142\/S0218195905001828","journal-title":"Int. J. Comput. Geom. Appl."},{"key":"503_CR4","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1023\/A:1005298816467","volume":"81","author":"Z F\u00fcredi","year":"2000","unstructured":"Z. F\u00fcredi, J.E. Wetzel, The smallest convex cover for triangles of perimeter two. Geom. Dedicata. 81, 285\u2013293 (2000). https:\/\/doi.org\/10.1023\/A:1005298816467","journal-title":"Geom. Dedicata."},{"key":"503_CR5","unstructured":"Gibbs, P.: An Upper bound for Lebesgue\u2019s Covering Problem (2018). arXiv preprint arXiv:1810.10089"},{"issue":"03","key":"503_CR6","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1142\/S0218195913500076","volume":"23","author":"T Khandhawit","year":"2013","unstructured":"T. Khandhawit, D. Pagonakis, S. Sriswasdi, Lower bound for convex hull area and universal cover problems. Int. J. Comput. Geom. Appl. 23(03), 197\u2013212 (2013). https:\/\/doi.org\/10.1142\/S0218195913500076","journal-title":"Int. J. Comput. Geom. Appl."},{"key":"503_CR7","first-page":"63","volume":"26","author":"MD Kovalev","year":"1983","unstructured":"M.D. Kovalev, A minimal convex covering for triangles (in Russian). Ukrain. Geom. Sb. 26, 63\u201368 (1983)","journal-title":"Ukrain. 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