Empty non-convex and convex four-gons in random point sets

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 2 logn +o(n 2 logn) and the expected number of empty convex four-gons with

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