Dual area measures and local additive kinematic formulas

Abstract

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natural convolution product which encodes the local additive kinematic formulas for groups acting transitively on the unit sphere. As an application of this new integral-geometric structure, we obtain the local additive kinematic formulas in hermitian vector spaces in a very explicit way.

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Acknowledgements

I thank Gil Solanes for many useful comments on a first version of this paper.

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Correspondence to Andreas Bernig.

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Supported by DFG Grant BE 2484/5-2.

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Bernig, A. Dual area measures and local additive kinematic formulas. Geom Dedicata 203, 85–110 (2019). https://doi.org/10.1007/s10711-019-00427-3

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Keywords

  • Area measure
  • Valuation
  • Kinematic formula
  • Integral geometry

Mathematics Subject Classification (2000)

  • 53C65
  • 52A22