  A polarized variety is K-stable if, for any test configuration, the
Donaldson-Futaki invariant is positive. In this paper, inspired by classical
geometric invariant theory, we describe the space of test configurations as a
limit of a direct system of Tits buildings. We show that the Donaldson-Futaki
invariant, conveniently normalized, is a continuous function on this space. We
also introduce a pseudo-metric on the space of test configurations. Recall that
K-stability can be enhanced by requiring that the Donaldson-Futaki invariant is
positive on any admissible filtration of the co-ordinate ring. We show that
admissible filtrations give rise to Cauchy sequences of test configurations
with respect to the above mentioned pseudo-metric.
