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.