A continuous analogue of the Upper Bound Theorem Wagner, Uli ; https://orcid.org/0000-0002-1494-0568 Welzl, Emo 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. Springer 2001 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/2419 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-001-0028-9 info:eu-repo/semantics/altIdentifier/issn/0179-5376 info:eu-repo/semantics/closedAccess