# Averaging Points Two at a Time

# Averaging Points Two at a Time

In 2006 Brendan McKay asked the following on sci.math.research: We have n points in a disk centered at the centroid of the points. We successively replace the two furthest points from each other by two copies of their average. (After each move we still have n points with the same centroid. How many moves are necessary to guarantee that all points lie in the concentric disk of half the radius?This really is the wrong question: it turns out that the situation is easier to study of we use a general Euclidean space and look at the rate of decay of the diameter in terms of number of moves. We get sharp asymptotic upper and lower bounds on the maximum diameter after certain numbers of moves. This involves interesting geometrical configurations and simple linear-programming arguments.