H\"older continuity for support measures of convex bodies

TL;DR

This paper establishes a H"older continuity estimate for support measures of convex bodies, improving the understanding of their stability and convergence properties in geometric measure theory.

Contribution

It provides a quantitative H"older estimate for support measures in the bounded Lipschitz metric, enhancing stability results for convex bodies and their area measures.

Findings

Support measures are weakly continuous with a H"older estimate.

The result offers a reverse stability estimate for Minkowski's existence theorem.

Improves understanding of geometric stability in convex analysis.

Abstract

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.