10.1145/336154.336176
Uli Wagner
Uli
Wagner0000-0002-1494-0568
Welzl, Emo
Emo
Welzl
Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem
ACM
2000
2018-12-11T11:57:33Z
2019-04-26T07:22:12Z
conference
https://research-explorer.app.ist.ac.at/record/2418
https://research-explorer.app.ist.ac.at/record/2418.json
For an absolutely continuous probability measure μ on Rd and a nonnegative integer k, let sk(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 sk(μ, 0), and we characterize the measures in Rd which attain this bound. 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 for simplicial convex polytopes.