# A continuous analogue of the Upper Bound Theorem

Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. 26(2), 205–219.

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Wagner, Uli

^{IST Austria}^{}; Welzl, EmoAbstract

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.

Publishing Year

Date Published

2001-01-01

Journal Title

Discrete & Computational Geometry

Volume

26

Issue

2

Page

205 - 219

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### Cite this

Wagner U, Welzl E. A continuous analogue of the Upper Bound Theorem.

*Discrete & Computational Geometry*. 2001;26(2):205-219. doi:10.1007/s00454-001-0028-9Wagner, U., & Welzl, E. (2001). A continuous analogue of the Upper Bound Theorem.

*Discrete & Computational Geometry*. Springer. https://doi.org/10.1007/s00454-001-0028-9Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.”

*Discrete & Computational Geometry*. Springer, 2001. https://doi.org/10.1007/s00454-001-0028-9.U. Wagner and E. Welzl, “A continuous analogue of the Upper Bound Theorem,”

*Discrete & Computational Geometry*, vol. 26, no. 2. Springer, pp. 205–219, 2001.Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.”

*Discrete & Computational Geometry*, vol. 26, no. 2, Springer, 2001, pp. 205–19, doi:10.1007/s00454-001-0028-9.