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Tight bounds on the expected number of holes in random point sets

RANDOM STRUCTURES & ALGORITHMS (2023)

Holes and islands in random point sets

Martin Balko et al.

Summary: This text discusses the concepts of k-islands and k-holes in a set of points in general position (S), as well as the expected number of k-holes in S when points are randomly chosen from a convex body K with volume 1. It shows that the expected number of k-holes in S is in O(nd) and provides an improved and generalized estimate compared to previous bounds. Additionally, it estimates the expected number of empty simplices in S and gives an asymptotically tight upper bound O(nd) even in a more general setting.

RANDOM STRUCTURES & ALGORITHMS (2022)

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