Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem

U. Wagner, E. Welzl, in:, ACM, 2000, pp. 50–56.

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Abstract
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.
Publishing Year
Date Published
2000-06-01
Page
50 - 56
Conference
SCG: Symposium on Computational Geometry
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Wagner U, Welzl E. Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. In: ACM; 2000:50-56. doi:10.1145/336154.336176
Wagner, U., & Welzl, E. (2000). Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem (pp. 50–56). Presented at the SCG: Symposium on Computational Geometry, ACM. https://doi.org/10.1145/336154.336176
Wagner, Uli, and Emo Welzl. “Origin-Embracing Distributions or a Continuous Analogue of the Upper Bound Theorem,” 50–56. ACM, 2000. https://doi.org/10.1145/336154.336176.
U. Wagner and E. Welzl, “Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem,” presented at the SCG: Symposium on Computational Geometry, 2000, pp. 50–56.
Wagner U, Welzl E. 2000. Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. SCG: Symposium on Computational Geometry 50–56.
Wagner, Uli, and Emo Welzl. Origin-Embracing Distributions or a Continuous Analogue of the Upper Bound Theorem. ACM, 2000, pp. 50–56, doi:10.1145/336154.336176.

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