Crofton Formulae for Tensor-Valued Curvature Measures

被引:4
|
作者
Hug, Daniel [1 ]
Weis, Jan A. [1 ]
机构
[1] Karlsruhe Inst Technol, Dept Math, Englerstr 2, D-76131 Karlsruhe, Germany
来源
TENSOR VALUATIONS AND THEIR APPLICATIONS IN STOCHASTIC GEOMETRY AND IMAGING | 2017年 / 2177卷
关键词
D O I
10.1007/978-3-319-51951-7_5
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a set of Crofton formulae for such tensorial curvature measures. These formulae express the integral mean of the tensorial curvature measures of the intersection of a given convex body with a uniform affine k-flat in terms of linear combinations of tensorial curvature measures of the given convex body. Here we first focus on the case where the tensorial curvature measures of the intersection of the given body with an affine flat is defined with respect to the affine flat as its ambient space. From these formulae we then deduce some new and also recover known special cases. In particular, we substantially simplify some of the constants that were obtained in previous work on Minkowski tensors. In a second step, we explain how the results can be extended to the case where the tensorial curvature measure of the intersection of the given body with an affine flat is determined with respect to the ambient Euclidean space.
引用
收藏
页码:111 / 156
页数:46
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