{"user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","page":"205 - 219","intvolume":"        26","publist_id":"4506","language":[{"iso":"eng"}],"quality_controlled":"1","author":[{"first_name":"Uli","last_name":"Wagner","full_name":"Wagner, Uli","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Welzl","full_name":"Welzl, Emo","first_name":"Emo"}],"doi":"10.1007/s00454-001-0028-9","_id":"2419","volume":26,"date_published":"2001-01-01T00:00:00Z","day":"01","publication_status":"published","publication":"Discrete & Computational Geometry","status":"public","extern":"1","article_type":"original","oa_version":"None","abstract":[{"text":"For an absolutely continuous probability measure μ. on ℝd and a nonnegative integer k, let S̃k(μ, 0) denote the probability that the convex hull of k + d + 1 random points which are i.i.d. according to μ contains the origin 0. For d and k given, we determine a tight upper bound on S̃k(μ, 0), and we characterize the measures in ℝd which attain this bound. As we will see, this result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex polytopes with a given number of vertices. For our proof we introduce so-called h-functions, continuous counterparts of h-vectors of simplicial convex polytopes.","lang":"eng"}],"year":"2001","publication_identifier":{"issn":["0179-5376"]},"issue":"2","publisher":"Springer","article_processing_charge":"No","scopus_import":"1","date_updated":"2023-05-24T13:13:51Z","month":"01","title":"A continuous analogue of the Upper Bound Theorem","acknowledgement":"We are indebted to Rolf Schneider for many helpful remarks and in particular for bringing reference [6] to our attention","citation":{"chicago":"Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2001. <a href=\"https://doi.org/10.1007/s00454-001-0028-9\">https://doi.org/10.1007/s00454-001-0028-9</a>.","apa":"Wagner, U., &#38; Welzl, E. (2001). A continuous analogue of the Upper Bound Theorem. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-001-0028-9\">https://doi.org/10.1007/s00454-001-0028-9</a>","short":"U. Wagner, E. Welzl, Discrete &#38; Computational Geometry 26 (2001) 205–219.","ista":"Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem. Discrete &#38; Computational Geometry. 26(2), 205–219.","ieee":"U. Wagner and E. Welzl, “A continuous analogue of the Upper Bound Theorem,” <i>Discrete &#38; Computational Geometry</i>, vol. 26, no. 2. Springer, pp. 205–219, 2001.","ama":"Wagner U, Welzl E. A continuous analogue of the Upper Bound Theorem. <i>Discrete &#38; Computational Geometry</i>. 2001;26(2):205-219. doi:<a href=\"https://doi.org/10.1007/s00454-001-0028-9\">10.1007/s00454-001-0028-9</a>","mla":"Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.” <i>Discrete &#38; Computational Geometry</i>, vol. 26, no. 2, Springer, 2001, pp. 205–19, doi:<a href=\"https://doi.org/10.1007/s00454-001-0028-9\">10.1007/s00454-001-0028-9</a>."},"type":"journal_article","date_created":"2018-12-11T11:57:33Z"}