{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:27:57Z","timestamp":1778495277297,"version":"3.51.4"},"reference-count":5,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5397,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Consider the hypercube [0, 1]<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> in <jats:italic>R<jats:sup>n<\/jats:sup><\/jats:italic>. This has 2<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> vertices and volume 1. Pick <jats:italic>N<\/jats:italic> = <jats:italic>N(n)<\/jats:italic> vertices independently at random, form their convex hull, and let <jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic> be its expected volume. How large should <jats:italic>N(n)<\/jats:italic> be to pick up significant volume? Let <jats:italic>k<\/jats:italic>=2\/\u221a\u22481.213, and let \u03f5 &gt; 0. We shall show that, as n\u2192\u221e, <jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic>\u21920 if N(n)\u2a7d(k\u2212\u03f5)<jats:sup>n<\/jats:sup> \u21921 if <jats:italic>N(n)<\/jats:italic> \u2a7e (<jats:italic>k<\/jats:italic> + \u03f5)<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>. A similar result holds for sampling uniformly from within the hypercube, with constant <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nust001.gif\" xlink:title=\"chemical structure image\"\/>.<\/jats:p>","DOI":"10.1002\/rsa.3240030107","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T18:16:58Z","timestamp":1180203418000},"page":"91-106","source":"Crossref","is-referenced-by-count":27,"title":["Volumes Spanned by Random Points in the Hypercube"],"prefix":"10.1002","volume":"3","author":[{"given":"M. E.","family":"Dyer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Z.","family":"F\u00fcredi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"McDiarmid","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.2307\/2047241"},{"key":"e_1_2_1_3_2","volume-title":"CBMS Regional Conference Series in Applied Mathematics","author":"Bahadur R. R."},{"key":"e_1_2_1_4_2","first-page":"33","volume-title":"Random volumes in the n\u2010cube, in Polyhedral Combinatorics","author":"Dyer M. E.","year":"1990"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187701"},{"key":"e_1_2_1_6_2","volume-title":"An Introduction to Probability Theory and Its Applications","author":"Feller W.","year":"1971"}],"container-title":["Random Structures &amp; Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Frsa.3240030107","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/rsa.3240030107","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,17]],"date-time":"2023-10-17T00:47:12Z","timestamp":1697503632000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/rsa.3240030107"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,1]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,1]]}},"alternative-id":["10.1002\/rsa.3240030107"],"URL":"https:\/\/doi.org\/10.1002\/rsa.3240030107","archive":["Portico"],"relation":{},"ISSN":["1042-9832","1098-2418"],"issn-type":[{"value":"1042-9832","type":"print"},{"value":"1098-2418","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,1]]}}}