  We introduce a norm on the space of test configurations, which we call the
minimum norm. We conjecture that uniform K-stability with respect to this norm
is equivalent to the existence of a constant scalar curvature K\"ahler metric.
This notion of uniform K-stability is analogous to coercivity of the Mabuchi
functional. We characterise the triviality of test configurations, by showing
that a test configuration has zero minimum norm if and only if it has zero
$L^2$-norm, if and only if it is almost trivial.
  We prove that the existence of a twisted constant scalar curvature K\"ahler
metric implies uniform twisted K-stability with respect to the minimum norm,
when the twisting is ample.
  We give algebro-geometric proofs of uniform K-stability in the general type
and Calabi-Yau cases, as well as in the Fano case under an alpha invariant
condition. Our results hold for line bundles sufficiently close to the
(anti)-canonical line bundle, and also in the twisted setting. We show that log
K-stability implies twisted K-stability, and also that twisted K-semistability
of a variety implies that the variety has mild singularities.
