Hug, D. & Schneider, R. Arch. Math. (2015) 104: 83. doi:10.1007/s00013-014-0719-0
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 Hölder 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.
Support measure Curvature measure Area measure Weak convergence Hölder continuity Stability