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 12 n(2) log n + o(n(2) log n) and the expected number of empty convex four-gons with vertices from S is Theta(n(2)).
Fabila, R.; Huemer, C.; Mitsche, D. Empty non-convex and convex four-gons in random point sets. "Studia scientiarum mathematicarum hungarica", 01 Març 2015, vol. 52, núm. 1, p. 52-64.