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Convergence of total variation of curvature measures

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Abstract.

It is known that the curvature measures of parallel ɛ-neighbourhoods of a set with positive reach or a polyconvex set converge vaguely if ɛ 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.

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References

  • H Federer (1959) ArticleTitleCurvature measures Trans Amer Math Soc 93 418–491 Occurrence Handle0089.38402 Occurrence Handle10.2307/1993504 Occurrence Handle110078

    Article  MATH  MathSciNet  Google Scholar 

  • H Federer (1969) Geometric Measure Theory Springer Berlin Heidelberg New York Occurrence Handle0176.00801

    MATH  Google Scholar 

  • G Matheron (1975) Random Sets and Integral Geometry Wiley New York Occurrence Handle0321.60009

    MATH  Google Scholar 

  • J Rataj (2004) ArticleTitleOn estimation of the Euler number by projections of thin slabs Adv Appl Prob 36 715–724 Occurrence Handle1070.60010 Occurrence Handle10.1239/aap/1093962230 Occurrence Handle2079910

    Article  MATH  MathSciNet  Google Scholar 

  • J Rataj (2006) ArticleTitleEstimation of intrinsic volumes from parallel neighbourhoods Suppl Rend Circ Palermo. Serie II 77 553–563 Occurrence Handle2245722

    MathSciNet  Google Scholar 

  • J Rataj M Zähle (2001) ArticleTitleCurvatures and currents for unions of sets with positive reach, II Ann Global Anal Geom 20 1–21 Occurrence Handle0997.53062 Occurrence Handle10.1023/A:1010624214933 Occurrence Handle1846894

    Article  MATH  MathSciNet  Google Scholar 

  • R Schneider (1980) ArticleTitleParallelmengen mit Vielfachheit und Steiner-Formeln Geom Dedicata 9 111–127 Occurrence Handle0435.52004 Occurrence Handle10.1007/BF00156479 Occurrence Handle566443

    Article  MATH  MathSciNet  Google Scholar 

  • R Schneider (1993) Convex Bodies: The Brunn-Minkowski Theory Univ Press Cambridge Occurrence Handle0798.52001

    MATH  Google Scholar 

  • Winter S (2006) Curvature Measures and Fractals. Thesis, Friedrich-Schiller-University Jena

  • M Zähle (1986) ArticleTitleIntegral and current representation of Federer’s curvature measures Arch Math 46 557–567 Occurrence Handle0598.53058 Occurrence Handle10.1007/BF01195026

    Article  MATH  Google Scholar 

  • M Zähle (1987) ArticleTitleCurvatures and currents for unions of sets with positive reach Geom Dedicata 23 155–171 Occurrence Handle0627.53053 Occurrence Handle10.1007/BF00181273 Occurrence Handle892398

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Charles University, Prague, Czech Republic

    J. Rataj

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  1. J. Rataj
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Correspondence to J. Rataj.

Additional information

Supported by MSM 113200007 and GAČR 201/06/0302.

Author’s address: Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

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Rataj, J. Convergence of total variation of curvature measures. Monatsh Math 153, 153–164 (2008). https://doi.org/10.1007/s00605-007-0492-2

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  • DOI: https://doi.org/10.1007/s00605-007-0492-2

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