A continuous analogue of the Upper Bound Theorem

U. Wagner, E. Welzl, Discrete & Computational Geometry 26 (2001) 205–219.

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Abstract
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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Wagner U, Welzl E. A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. 2001;26(2):205-219. doi:10.1007/s00454-001-0028-9
Wagner, U., & Welzl, E. (2001). A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry, 26(2), 205–219. https://doi.org/10.1007/s00454-001-0028-9
Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.” Discrete & Computational Geometry 26, no. 2 (2001): 205–19. 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, pp. 205–219, 2001.
Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. 26(2), 205–219.
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.

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