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

Wagner U, Welzl E. 2000. Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. Proceedings of the 16th annual symposium on Computational geometry. SCG: Symposium on Computational Geometry, 50–56.

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Author

Wagner, Uli

^{ISTA}^{}; Welzl, EmoAbstract

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

2000-05-01

Proceedings Title

Proceedings of the 16th annual symposium on Computational geometry

Page

50 - 56

Conference

SCG: Symposium on Computational Geometry

Conference Location

Clear Water Bay Kowloon, Hong Kong

Conference Date

2000-06-12 – 2000-04-14

ISBN

IST-REx-ID

### Cite this

Wagner U, Welzl E. Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. In:

*Proceedings of the 16th Annual Symposium on Computational Geometry*. ACM; 2000:50-56. doi:10.1145/336154.336176Wagner, U., & Welzl, E. (2000). Origin-embracing distributions or a continuous analogue of the Upper Bound Theorem. In

*Proceedings of the 16th annual symposium on Computational geometry*(pp. 50–56). Clear Water Bay Kowloon, Hong Kong: ACM. https://doi.org/10.1145/336154.336176Wagner, Uli, and Emo Welzl. “Origin-Embracing Distributions or a Continuous Analogue of the Upper Bound Theorem.” In

*Proceedings of the 16th Annual Symposium on Computational Geometry*, 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,” in

*Proceedings of the 16th annual symposium on Computational geometry*, Clear Water Bay Kowloon, Hong Kong, 2000, pp. 50–56.Wagner, Uli, and Emo Welzl. “Origin-Embracing Distributions or a Continuous Analogue of the Upper Bound Theorem.”

*Proceedings of the 16th Annual Symposium on Computational Geometry*, ACM, 2000, pp. 50–56, doi:10.1145/336154.336176.