Abstract
For an absolutely continuous probability measure μ on R d and a nonnegative integer k , let \tilde s k (μ ,\origin ) denote the probability that the convex hull of k+d+1 random points which are i.i.d. according to μ contains the origin \bf 0 . For d and k given, we determine a tight upper bound on \tilde s k (μ ,\origin ) , and we characterize the measures in R d 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.
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Received April 14, 2000, and in revised form October 6, 2000. Online publication June 20, 2001.
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Wagner, U., Welzl, E. A Continuous Analogue of the Upper Bound Theorem. Discrete Comput Geom 26, 205–219 (2001). https://doi.org/10.1007/s00454-001-0028-9
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DOI: https://doi.org/10.1007/s00454-001-0028-9