Prof. Miguel Mosteiro spoke on a problem regarding Delaunay triangulations and randomization:

Title: Probabilistic Bounds on the Length of a Longest Edge in Delaunay Graphs of Random Points in d-Dimensions

Abstract: Motivated by low energy consumption in geographic routing in wireless networks, there has been recent 
interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. 
Asymptotic results are known for random networks in planar domains. I will overview upper and lower bounds 
we obtained recently, that hold with parametric probability in any dimension, for points distributed uniformly 
at random in domains with and without boundary. These bounds are asymptotically tight for all relevant values 
of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in 
the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths 
of long edges in Delaunay graphs.

(Joint work with Estie Arkin, Antonio Fernandez-Anta, and Joe Mitchell.)