{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T08:01:03Z","timestamp":1787385663512,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>\n                    Heilbronn conjectured that given arbitrary n points from the 2-dimensional unit square, there must be three points which form a triangle of area at most O(1\/n\n                    <jats:sup>2<\/jats:sup>\n                    ). This conjecture was disproved by a nonconstructive argument of Koml\u00f3s, Pintz, and Szemer\u00e9di [ J. London Math. Soc., 25 (1982), pp. 13--24] who showed that for every n there is a configuration of n points in the unit square where all triangles have area at least $\\Omega({\\log n}\/{n^2})$. Considering a discretization of Heilbronn's problem, we give an alternative proof of the result from [J. London Math. Soc., 25 (1982), pp. 13--24]. Our approach has two advantages: First, it yields a polynomial-time algorithm which for every n computes a configuration of n points where all triangles have area $\\Omega({\\log n}\/{n^2})$. Second, it allows us to consider a generalization of Heilbronn's problem to convex hulls of k points where we can show that an algorithmic solution is also available.\n                  <\/jats:p>","DOI":"10.1137\/s0097539798348870","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"383-390","source":"Crossref","is-referenced-by-count":19,"title":["An Algorithm for Heilbronn's Problem"],"prefix":"10.1137","volume":"30","author":[{"given":"Claudia","family":"Bertram--Kretzberg","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Hofmeister","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hanno","family":"Lefmann","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,27]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(82)90049-8"},{"key":"R2","unstructured":"Gill Barequet, A lower bound for Heilbronn\u2019s triangle problem in d dimensions, ACM, New York, 1999, 76\u2013811739933"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539797323716"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199603)8:2<131::AID-RSA4>3.0.CO;2-Z"},{"key":"R5","volume-title":"Handbook of combinatorics. Vol. 1, 2","author":"Graham R.","year":"1995"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"J. H\u00e5stad,\n                      Clique is hard to approximate within n1-\u03b5\n                      , in Proceedings of the 37th IEEE Symposium on the Foundations of Computer Science (FOCS), 1996, pp. 627\u2013636.","DOI":"10.1109\/SFCS.1996.548522"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"Thomas Hofmeister, Hanno Lefmann, Approximating maximum independent sets in uniform hypergraphs, Lecture Notes in Comput. Sci., Vol. 1450, Springer, Berlin, 1998, 562\u20135701684102","DOI":"10.1007\/BFb0055806"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-24.3.385"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-25.1.13"},{"key":"R10","unstructured":"H. Lefmann,\n                      On Heilbronn\u2019s problem in higher dimension\n                      , in Proceedings of the 11th Annual ACM\u2013SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, PA, 2000, pp. 60\u201364."},{"key":"R11","volume-title":"An introduction to the theory of numbers","author":"Niven Ivan","year":"1991"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-26.3.198"},{"key":"R13","first-page":"0","volume":"25","author":"Roth K.","year":"1972","journal-title":"Proc. London Math. Soc. (3)"},{"key":"R14","first-page":"0","volume":"25","author":"Roth K.","year":"1972","journal-title":"Proc. London Math. Soc. (3)"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"K. Roth, Estimation of the area of the smallest triangle obtained by selecting three out of n points in a disc of unit area, Amer. Math. Soc., Providence, R.I., 1973, 251\u201326249:226","DOI":"10.1090\/pspum\/024\/0335445"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(76)90100-6"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-4.3.545"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(72)90084-2"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539798348870","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:21:52Z","timestamp":1787336512000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539798348870"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S0097539798348870"],"URL":"https:\/\/doi.org\/10.1137\/s0097539798348870","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}