Integral geometry of pairs of hyperplanes or lines

Author:

Hug Daniel,Schneider Rolf

Abstract

AbstractCrofton’s formula of integral geometry evaluates the total motion invariant measure of the set of k-dimensional planes having nonempty intersection with a given convex body. This note deals with motion invariant measures on sets of pairs of hyperplanes or lines meeting a convex body. Particularly simple results are obtained if, and only if, the given body is of constant width in the first case, and of constant brightness in the second case.

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics

Reference9 articles.

1. Crofton, M.W.: On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus. Philos. Trans. R. Soc. Lond. 158, 181–199 (1868)

2. Cufí, J., Gallego, E., Reventós, A.: Integral geometry about the visual angle of a convex set. arXiv:1906.10374v1

3. Gardner, R.J.: Geometric Tomography. 2nd edn., Encyclopedia of Mathematics and Its Applications, vol. 58. Cambridge University Press, New York (2006)

4. Kallenberg, O.: Invariant measures and disintegrations with applications to Palm and related kernels. Probab. Theory Relat. Fields 139, 285–310 (2007). Erratum: Probab. Theory Relat. Fields 139, 311 (2007)

5. Martini, H., Montejano, L., Oliveros, D.: Bodies of Constant Width. Birkhäuser/Springer, Cham (2019)

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Integral Geometry of Pairs of Lines and Planes;New Trends in Geometric Analysis;2023

2. Integral geometry of pairs of planes;Archiv der Mathematik;2021-09-21

3. Correction To: Integral geometry of pairs of hyperplanes or lines;Archiv der Mathematik;2021-09-16

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