Convergence of total variation of curvature measures. (English) Zbl 1136.53054
Summary: It is known that the curvature measures of parallel \(\varepsilon\)-neighbourhoods of a set with positive reach or a polyconvex set converge vaguely if \(\varepsilon\) tends to zero to the curvature measures of the set itself. We show that in the case of a set with positive reach, the total variations of the curvature measures converge as well, whereas in the case of a polyconvex set this is no more true in general.
MSC:
| 53C65 | Integral geometry |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
Keywords:
curvature measures; set with positive reach; total variation of signed measures; weak convergenceReferences:
| [9] | Winter S (2006) Curvature Measures and Fractals. Thesis, Friedrich-Schiller-University Jena · Zbl 1115.53309 |
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