Hölder Continuity of Normal Cycles and of Support Measures of Convex Bodies

The normal cycle TK associated with a convex body K ⊂ Rn is a current which in principle contains complete information about K. It is known that if a sequence of convex bodies Ki, i ∈ N, converges to a convex body K in the Hausdorff metric, then the associated normal cycles TKi converge to TK in the dual flat seminorm. We give a quantitative improvement of this convergence result by providing an estimate of the distance (in the dual flat seminorm) of the normal cycles of convex bodies with given Hausdorff distance. The support measures of a convex body K arise from a local Steiner formula or, alternatively, by evaluating suitable differential forms at the normal cycle of K. Complementing the estimate for the normal cycles, we establish an upper bound for the distance (in the bounded Lipschitz metric) of the support measures of two convex bodies in terms of the Hausdorff distance of these bodies. A special case of these estimates yields reverse forms of known stability results for area measures.