  The support measures of a convex body are a common generalization of the
curvature measures and the area measures. With respect to the Hausdorff metric
on the space of convex bodies, they are weakly continuous. We provide a
quantitative improvement of this result, by establishing a H\"older estimate
for the support measures in terms of the bounded Lipschitz metric, which
metrizes the weak convergence. Specializing the result to area measures yields
a reverse counterpart to earlier stability estimates, concerning Minkowski's
existence theorem for convex bodies with given area measure.
