Archiv der Mathematik

, Volume 104, Issue 1, pp 83–92

Hölder continuity for support measures of convex bodies


DOI: 10.1007/s00013-014-0719-0

Cite this article as:
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 

Mathematics Subject Classification

Primary 52A20 Secondary 52A22 

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Mathematisches InstitutAlbert-Ludwigs-UniversitätFreiburgGermany

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