Hölder continuity for support measures of convex bodies
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.
KeywordsSupport measure Curvature measure Area measure Weak convergence Hölder continuity Stability
Mathematics Subject ClassificationPrimary 52A20 Secondary 52A22
Unable to display preview. Download preview PDF.
- 2.V. Diskant, Bounds for the discrepancy between convex bodies in terms of the isoperimetric difference (in Russian), Sibirskii Mat. Zh. 13 (1972), 767–772. English translation: Siberian Math. J. 13 (1973), 529–532.Google Scholar
- 3.R. M. Dudley, Real Analysis and Probability. Cambridge University Press, New York, 2002.Google Scholar
- 4.H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491.Google Scholar
- 5.H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.Google Scholar
- 6.P. Goodey, M. Kiderlen, and W. Weil, Spherical projections and liftings in geometric tomography, Adv. Geom. 11 (2011), 1–47.Google Scholar
- 7.D. Hug and R. Schneider, Stability results involving surface area measures of convex bodies, Rend. Circ. Mat. Palermo (2) Suppl. 70 (2002), vol. II, 21–51.Google Scholar
- 8.R. Schneider, Convex Bodies: The Brunn–Minkowski Theory. 2nd. ed., Cambridge University Press, Cambridge, 2014.Google Scholar