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distances IEEE Trans. Knowl. Data Eng. 19 873 – 886 10.1109/TKDE.2007.1037 . [8] Gates , D. 1985 Asymptotics of two integrals from optimization theory and geometric

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Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n 2logn + o(n 2logn) and the expected number of empty convex four-gons with vertices from S is Θ(n 2).

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Santaló, L. A. , Integral Geometry and Geometric Probability , Encyclopedia of Mathematics and its Applications, 1 , Addison-Wesley, Reading, Mass. etc., 1976, MR 55 #6340. Santaló L. A

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