{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T22:49:05Z","timestamp":1673304545847},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2011,1,28]],"date-time":"2011-01-28T00:00:00Z","timestamp":1296172800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,7]]},"abstract":"<jats:p>Let <jats:italic>P<\/jats:italic> be a set of <jats:italic>n<\/jats:italic> points in \u211d<jats:sup>3<\/jats:sup>, and let <jats:italic>k<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic> be an integer. A sphere \u03c3 is <jats:italic>k-rich<\/jats:italic> with respect to <jats:italic>P<\/jats:italic> if |\u03c3 \u2229 <jats:italic>P<\/jats:italic>| \u2265 <jats:italic>k<\/jats:italic>, and is \u03b7-<jats:italic>non-degenerate<\/jats:italic>, for a fixed fraction 0 &lt; \u03b7 &lt; 1, if no circle \u03b3 \u2282 \u03c3 contains more than \u03b7|\u03c3 \u2229 <jats:italic>P<\/jats:italic>| points of <jats:italic>P<\/jats:italic>.<\/jats:p><jats:p>We improve the previous bound given in [1] on the number of <jats:italic>k<\/jats:italic>-rich \u03b7-non-degenerate spheres in 3-space with respect to any set of <jats:italic>n<\/jats:italic> points in \u211d<jats:sup>3<\/jats:sup>, from <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>4<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>5<\/jats:sup> + <jats:italic>n<\/jats:italic><jats:sup>3<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>3<\/jats:sup>), which holds for all 0 &lt; \u03b7 &lt; 1\/2, to <jats:italic>O<\/jats:italic>*(<jats:italic>n<\/jats:italic><jats:sup>4<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>11\/2<\/jats:sup> + <jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>\/<jats:italic>k<\/jats:italic><jats:sup>2<\/jats:sup>), which holds for all 0 &lt; \u03b7 &lt; 1 (in both bounds, the constants of proportionality depend on \u03b7). The new bound implies the improved upper bound <jats:italic>O<\/jats:italic>*(<jats:italic>n<\/jats:italic><jats:sup>58\/27<\/jats:sup>) \u2248 <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2.1482<\/jats:sup>) on the number of mutually similar triangles spanned by <jats:italic>n<\/jats:italic> points in \u211d<jats:sup>3<\/jats:sup>; the previous bound was <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>13\/6<\/jats:sup>) \u2248 <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2.1667<\/jats:sup>) [1].<\/jats:p>","DOI":"10.1017\/s0963548311000010","type":"journal-article","created":{"date-parts":[[2011,1,28]],"date-time":"2011-01-28T12:36:25Z","timestamp":1296218185000},"page":"503-512","source":"Crossref","is-referenced-by-count":5,"title":["Non-Degenerate Spheres in Three Dimensions"],"prefix":"10.1017","volume":"20","author":[{"given":"ROEL","family":"APFELBAUM","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MICHA","family":"SHARIR","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,1,28]]},"reference":[{"key":"S0963548311000010_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-004-1111-9"},{"key":"S0963548311000010_ref9","doi-asserted-by":"crossref","unstructured":"[9] Elekes G. and T\u00f3th C. D. (2005) Incidences of not too degenerate hyperplanes. In Proc. 21st Annu. ACM Sympos. Comput. Geom., pp. 16\u201321.","DOI":"10.1145\/1064092.1064098"},{"key":"S0963548311000010_ref1","doi-asserted-by":"crossref","unstructured":"[1] Agarwal P. K. , Apfelbaum R. , Purdy G. and Sharir M. (2007) Similar simplices in a d-dimensional point set. In Proc. 23rd Annu. ACM Sympos. Comput. Geom., pp. 232\u2013238.","DOI":"10.1145\/1247069.1247112"},{"key":"S0963548311000010_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2005.07.002"},{"key":"S0963548311000010_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61568-9"},{"key":"S0963548311000010_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548304006091"},{"key":"S0963548311000010_ref6","volume-title":"Handbook of Data Structures and Applications","author":"Chazelle","year":"2005"},{"key":"S0963548311000010_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-008-2099-1"},{"key":"S0963548311000010_ref2","doi-asserted-by":"publisher","DOI":"10.1145\/972639.972641"},{"key":"S0963548311000010_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187783"},{"key":"S0963548311000010_ref10","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/342\/06136"},{"key":"S0963548311000010_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-001-0084-1"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000010","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T07:57:10Z","timestamp":1556351830000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000010\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,28]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,7]]}},"alternative-id":["S0963548311000010"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000010","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,28]]}}}