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Crofton Formulae for Tensorial Curvature Measures: The General Case

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Analytic Aspects of Convexity

Part of the book series: Springer INdAM Series ((SINDAMS,volume 25))

Abstract

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. On convex polytopes, there exist further generalizations some of which also have continuous extensions to arbitrary convex bodies. In a previous work, we obtained kinematic formulae for all (generalized) tensorial curvature measures. As a consequence of these results, we now derive a complete system of Crofton formulae for such (generalized) tensorial curvature measures. These formulae express the integral mean of the (generalized) tensorial curvature measures of the intersection of a given convex body (resp. polytope, or finite unions thereof) with a uniform affine k-flat in terms of linear combinations of (generalized) tensorial curvature measures of the given convex body (resp. polytope, or finite unions thereof). The considered generalized tensorial curvature measures generalize those studied formerly in the context of Crofton-type formulae, and the coefficients involved in these results are substantially less technical and structurally more transparent than in previous works. Finally, we prove that essentially all generalized tensorial curvature measures on convex polytopes are linearly independent. In particular, this implies that the Crofton formulae which we prove in this contribution cannot be simplified further.

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Acknowledgements

The authors were supported in part by DFG grants FOR 1548.

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Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Daniel Hug & Jan A. Weis

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  1. Daniel Hug
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  2. Jan A. Weis
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Correspondence to Daniel Hug .

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Editors and Affiliations

  1. Dept. of Mathematics and Computer Science “U. Dini”, University of Florence, Florence, Italy

    Gabriele Bianchi

  2. Dept. of Mathematics and Computer Science “U. Dini”, University of Florence, Florence, Italy

    Andrea Colesanti

  3. Dept. of Mathematics and Computer Science “U. Dini”, University of Florence, Florence, Italy

    Paolo Gronchi

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Hug, D., Weis, J.A. (2018). Crofton Formulae for Tensorial Curvature Measures: The General Case. In: Bianchi, G., Colesanti, A., Gronchi, P. (eds) Analytic Aspects of Convexity. Springer INdAM Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-71834-7_3

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