  We study the asymptotics of a Markovian system of $N \geq 3$ particles in
$[0,1]^d$ in which, at each step in discrete time, the particle farthest from
the current centre of mass is removed and replaced by an independent $U
[0,1]^d$ random particle. We show that the limiting configuration contains
$N-1$ coincident particles at a random location $\xi_N \in [0,1]^d$. A key tool
in the analysis is a Lyapunov function based on the squared radius of gyration
(sum of squared distances) of the points. For d=1 we give additional results on
the distribution of the limit $\xi_N$, showing, among other things, that it
gives positive probability to any nonempty interval subset of $[0,1]$, and
giving a reasonably explicit description in the smallest nontrivial case, N=3.
