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A flag representation of projection functions

  • Paul Goodey , Wolfram Hinderer , Daniel Hug , Jan Rataj and Wolfgang Weil EMAIL logo
Published/Copyright: July 22, 2017
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Advances in Geometry
From the journal Advances in Geometry Volume 17 Issue 3

Abstract

The kth projection function υk(K, ·) of a convex body K ⊂ ℝd, d ≥ 3, is a function on the Grassmannian G(d, k) which measures the k-dimensional volume of the projection of K onto members of G(d, k). For k = 1 and k = d − 1, simple formulas for the projection functions exist. In particular, υd-1(K, ·) can be written as a spherical integral with respect to the surface area measure of K. Here, we generalize this result and prove two integral representations for υk(K, ·), k = 1,..., d − 1, over flag manifolds. Whereas the first representation generalizes a result of Ambartzumian (1987), but uses a flag measure which is not continuous in K, the second representation is related to a recent flag formula for mixed volumes by Hug, Rataj and Weil (2013) and depends continuously on K.

Keywords: Projection functions; intrinsic volumes; flag measures; convex bodies; Grassmannian; generalized curvatures
MSC 2010: 52A20; 52A22; 52A39; 53C65

M. Henk

Acknowledgements

The proof of Theorem 7.2 is based on a joint work of DH with Manfred Peter.

Funding

The research of DH and WW has been supported by DFG projects HU 1874/4-2 and WE 1613/2-2 and JR has been partially supported by grant GAČR P201/10/0472.

References

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Received: 2015-2-24
Revised: 2015-7-24
Published Online: 2017-7-22
Published in Print: 2017-7-26

© 2017 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  3. Classification of 3-dimensional left-invariant almost paracontact metric structures
  5. On the topology of the spaces of curvature constrained plane curves
  7. On deformations of parallel G2 structures and almost contact metric structures
  9. A flag representation of projection functions
  11. A fundamental theorem for submanifolds of multiproducts of real space forms
  13. Intriguing sets of quadrics in PG(5, q)
  15. Successive radii and ball operators in generalized Minkowski spaces
  17. Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
  19. The Archimedean projection property
  21. Kernels of numerical pushforwards
  23. Modified classical flat Minkowski planes
https://doi.org/10.1515/advgeom-2017-0022
Keywords for this article
Projection functions; intrinsic volumes; flag measures; convex bodies; Grassmannian; generalized curvatures

Articles in the same Issue

  1. Frontmatter
  3. Classification of 3-dimensional left-invariant almost paracontact metric structures
  5. On the topology of the spaces of curvature constrained plane curves
  7. On deformations of parallel G2 structures and almost contact metric structures
  9. A flag representation of projection functions
  11. A fundamental theorem for submanifolds of multiproducts of real space forms
  13. Intriguing sets of quadrics in PG(5, q)
  15. Successive radii and ball operators in generalized Minkowski spaces
  17. Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
  19. The Archimedean projection property
  21. Kernels of numerical pushforwards
  23. Modified classical flat Minkowski planes
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