Dual area measures and local additive kinematic formulas

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 t...

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Bibliographic Details
Published in:Geometriae dedicata 2019-12, Vol.203 (1), p.85-110
Main Author: Bernig, Andreas
Format: Article
Language:English
Subjects:
Algebraic Geometry
Convex and Discrete Geometry
Convolution
Differential Geometry
Euclidean geometry
Hyperbolic Geometry
Kinematics
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Topology
Vector space
Vector spaces
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Summary: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.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-019-00427-3