A continuous analogue of the Upper Bound Theorem

article A continuous analogue of the Upper Bound Theorem published yes Uli Wagner author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568 Emo Welzl author 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. Springer2001 eng Discrete & Computational Geometry 0179-537610.1007/s00454-001-0028-9 262205 - 219 yes Wagner U, Welzl E. A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. 2001;26(2):205-219. doi:<a href="https://doi.org/10.1007/s00454-001-0028-9">10.1007/s00454-001-0028-9</a> Wagner, Uli, and Emo Welzl. “A Continuous Analogue of the Upper Bound Theorem.” Discrete & Computational Geometry. Springer, 2001. <a href="https://doi.org/10.1007/s00454-001-0028-9">https://doi.org/10.1007/s00454-001-0028-9</a>. 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:<a href="https://doi.org/10.1007/s00454-001-0028-9">10.1007/s00454-001-0028-9</a>. Wagner U, Welzl E. 2001. A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. 26(2), 205–219. Wagner, U., & Welzl, E. (2001). A continuous analogue of the Upper Bound Theorem. Discrete & Computational Geometry. Springer. <a href="https://doi.org/10.1007/s00454-001-0028-9">https://doi.org/10.1007/s00454-001-0028-9</a> U. Wagner, E. Welzl, Discrete & Computational Geometry 26 (2001) 205–219. 24192018-12-11T11:57:33Z2023-05-24T13:13:51Z