Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem
Uli Wagner
Welzl, Emo
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.
ACM
2000
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https://research-explorer.app.ist.ac.at/record/2418
Wagner U, Welzl E. Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. In: ACM; 2000:50-56. doi:<a href="https://doi.org/10.1145/336154.336176">10.1145/336154.336176</a>
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