  In this paper, we generalise results obtained earlier by John Cremona and the
author on the reduction theory of binary forms, which describe positive
zero-cycles in P^1, to positive zero-cycles (or point clusters) in projective
spaces of arbitrary dimension. This should have applications to more general
projective varieties in P^n, by associating a suitable positive zero-cycle to
them in an PGL(n+1)-invariant way. We discuss this in the case of (smooth)
plane curves.
