---
_id: '2419'
abstract:
- lang: eng
  text: 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.
acknowledgement: We are indebted to Rolf Schneider for many helpful remarks and in
  particular for bringing reference [6] to our attention
article_processing_charge: No
article_type: original
author:
- first_name: Uli
  full_name: Wagner, Uli
  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. A continuous analogue of the Upper Bound Theorem. <i>Discrete
    &#38; Computational Geometry</i>. 2001;26(2):205-219. doi:<a href="https://doi.org/10.1007/s00454-001-0028-9">10.1007/s00454-001-0028-9</a>
  apa: Wagner, U., &#38; Welzl, E. (2001). A continuous analogue of the Upper Bound
    Theorem. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href="https://doi.org/10.1007/s00454-001-0028-9">https://doi.org/10.1007/s00454-001-0028-9</a>
  chicago: Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.”
    <i>Discrete &#38; Computational Geometry</i>. Springer, 2001. <a href="https://doi.org/10.1007/s00454-001-0028-9">https://doi.org/10.1007/s00454-001-0028-9</a>.
  ieee: U. Wagner and E. Welzl, “A continuous analogue of the Upper Bound Theorem,”
    <i>Discrete &#38; Computational Geometry</i>, vol. 26, no. 2. Springer, pp. 205–219,
    2001.
  ista: Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem.
    Discrete &#38; Computational Geometry. 26(2), 205–219.
  mla: Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.”
    <i>Discrete &#38; Computational Geometry</i>, vol. 26, no. 2, Springer, 2001,
    pp. 205–19, doi:<a href="https://doi.org/10.1007/s00454-001-0028-9">10.1007/s00454-001-0028-9</a>.
  short: U. Wagner, E. Welzl, Discrete &#38; Computational Geometry 26 (2001) 205–219.
date_created: 2018-12-11T11:57:33Z
date_published: 2001-01-01T00:00:00Z
date_updated: 2023-05-24T13:13:51Z
day: '01'
doi: 10.1007/s00454-001-0028-9
extern: '1'
fulldoi: https://doi.org/10.1007/s00454-001-0028-9
intvolume: '        26'
issue: '2'
language:
- iso: eng
month: '01'
oa_version: None
page: 205 - 219
publication: Discrete & Computational Geometry
publication_identifier:
  issn:
  - 0179-5376
publication_status: published
publisher: Springer
publist_id: '4506'
quality_controlled: '1'
scopus_import: '1'
status: public
title: A continuous analogue of the Upper Bound Theorem
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 26
year: '2001'
...