[{"volume":26,"author":[{"orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","last_name":"Wagner","first_name":"Uli"},{"first_name":"Emo","last_name":"Welzl","full_name":"Welzl, Emo"}],"status":"public","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","year":"2001","page":"205 - 219","date_updated":"2023-05-24T13:13:51Z","language":[{"iso":"eng"}],"issue":"2","oa_version":"None","publisher":"Springer","date_created":"2018-12-11T11:57:33Z","title":"A continuous analogue of the Upper Bound Theorem","article_processing_charge":"No","scopus_import":"1","doi":"10.1007/s00454-001-0028-9","publication_identifier":{"issn":["0179-5376"]},"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>.","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>.","ista":"Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem. Discrete &#38; Computational Geometry. 26(2), 205–219.","short":"U. Wagner, E. Welzl, Discrete &#38; Computational Geometry 26 (2001) 205–219.","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>","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>"},"article_type":"original","date_published":"2001-01-01T00:00:00Z","intvolume":"        26","day":"01","type":"journal_article","fulldoi":"https://doi.org/10.1007/s00454-001-0028-9","publication_status":"published","_id":"2419","extern":"1","month":"01","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"}],"publication":"Discrete & Computational Geometry","publist_id":"4506","quality_controlled":"1","acknowledgement":"We are indebted to Rolf Schneider for many helpful remarks and in particular for bringing reference [6] to our attention"}]