Large convex holes in random point sets

Abstract

A convex hole (or empty convex polygon) of a point set P in the plane is a convex polygon with vertices in P , containing no points of P in its interior. Let R be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of n random points chosen independently and uniformly over R is Θ(log n/(log log n)), regardless of the shape of R.

Holes and islands in random point sets

“…This is in contrast with the fact that, unlike Horton sets, they contain arbitrarily large holes. Quite recently, Balogh, González-Aguilar, and Salazar [3] showed that the expected number of vertices of the largest hole in a set of n random points chosen independently and uniformly over a convex body in the plane is in Θ(log n/(log log n)).…”

“…FIGURE 5 An illustration of the proof of Lemma 10. In order for {p 1 , … , p i } to be an i-island in S, the light gray part cannot contain points from S. We estimate the probability of this event from above by the probability that the dark gray simplex conv(𝜑 ∪ {p i }) contains no point of S. Note that the parameters 𝜂 and 𝜏 coincide for 𝑑 = 2, as then…”

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

“…, for some constant K. Balogh et al [5] showed that the expected number of vertices of the largest empty convex polygon in S is Θ( log n log log n ).…”

Large Area Convex Holes in Random Point Sets

“…In [2], Balogh et al proved that w.h.p. the size of the largest (in the number of vertices) polygonal hole with vertices in K n is Θ(log n/ log log n).…”