---
_id: '2418'
abstract:
- lang: eng
text: 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.
author:
- first_name: Uli
full_name: Uli Wagner
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Emo
full_name: Welzl, Emo
last_name: Welzl
citation:
ama: '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'
apa: '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'
chicago: 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.
ieee: '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.'
ista: 'Wagner U, Welzl E. 2000. Origin-embracing distributions or a continuous analogue
of the Upper Bound Theorem. SCG: Symposium on Computational Geometry, 50–56.'
mla: 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.
short: U. Wagner, E. Welzl, in:, ACM, 2000, pp. 50–56.
conference:
name: 'SCG: Symposium on Computational Geometry'
date_created: 2018-12-11T11:57:33Z
date_published: 2000-06-01T00:00:00Z
date_updated: 2021-01-12T06:57:22Z
day: '01'
doi: 10.1145/336154.336176
extern: 1
month: '06'
page: 50 - 56
publication_status: published
publisher: ACM
publist_id: '4507'
quality_controlled: 0
status: public
title: Origin-embracing distributions or a continuous analogue of the Upper Bound
Theorem
type: conference
year: '2000'
...