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Mixed curvature measures of translative integral geometry

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Abstract

The curvature measures of a set X with singularities are measures concentrated on the normal bundle of X, which describe the local geometry of the set X. For given finitely many convex bodies or, more generally, sets with positive reach, the translative integral formula for curvature measures relates the integral mean of the curvature measures of the intersections of the given sets, one fixed and the others translated, to the mixed curvature measures of the given sets. In the case of two sets of positive reach, a representation of these mixed measures in terms of generalized curvatures, defined on the normal bundles of the sets, is known. For more than two sets, a description of mixed curvature measures in terms of rectifiable currents has been derived previously. Here we provide a representation of mixed curvature measures of sets with positive reach based on generalized curvatures. The special case of convex polyhedra is treated in detail.

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Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology (KIT), 76128, Karlsruhe, Germany

    Daniel Hug

  2. Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Jan Rataj

Authors
  1. Daniel Hug
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  2. Jan Rataj
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Correspondence to Daniel Hug.

Additional information

The research has been supported by the DFG Project HU 1874/4-2 (D. Hug) and the Czech Science Foundation, Project P201/15-08218S (J. Rataj).

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Hug, D., Rataj, J. Mixed curvature measures of translative integral geometry. Geom Dedicata 195, 101–120 (2018). https://doi.org/10.1007/s10711-017-0278-1

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  • DOI: https://doi.org/10.1007/s10711-017-0278-1

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