Crofton formulas and indefinite signature
Crofton formulas and indefinite signature
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Crofton 公式和不定签名
DOI：
10.1007/s00039-017-0406-y
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发表时间：
2016
影响因子：
2.2
通讯作者：
Dmitry Faifman
中科院分区：
文献类型：
--
作者：
Dmitry Faifman
摘要我们研究了A.Bernig和作者分类的O(p，q)-不变赋值。我们的主要结果是，每个这样的估值都是由O(p，q)-不变的Crofton公式给出的。这是通过首先获得几个足够普遍的签名和齐次程度的显式公式来实现的，特别是在O(p，p)的(p−1)齐次情况下，当$${p\NOT\EQUV3\mod时，得到中心仿射表面积的Crofton公式
4}$$p≢3mod4。然后，我们利用Crofton公式的函数性将其推广到一般情况。我们还明确地证明了所有O(p，2)-不变赋值的不变公式。证明依赖于一些独立的积分的精确计算。这些都与Selberg积分和矩阵变元的Beta函数有关，只是正定矩阵被所有签名的矩阵所取代。我们还分析了极小轨道上支撑的著名的不变Crofton分布，并表明，令人惊讶的是，它有时定义了平凡的赋值，从而产生了特别小支撑的余弦变换核中的分布。本文的核心是Muro对对称矩阵空间上的$$|deT X|^S}$$|DETX|S分布族的刻画，我们用它来构造一个O(p，q)不变的Crofton分布族。我们猜想没有其他的猜想，然后我们证明了对于P偶数的O(p，2)。T.Wannerer和作者在附录中研究了Crofton分布的函数性，这是我们研究的一个重要工具。
AbstractWe study the O(p, q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p, q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the (p − 1) homogeneous case of O(p, p), yielding a Crofton formula for the centro-affine surface area when $${p\not\equiv 3\mod
4}$$p≢3mod4. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p, 2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg’s integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the $${|\det X|^s}$$|detX|s family of distributions on the space of symmetric matrices, which we use to construct a family of O(p, q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p, 2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.
DOI：
10.1007/s00526-015-0843-0
发表时间：
2015-10-01
影响因子：
2.1
作者：
Abardia, Judit;Wannerer, Thomas
通讯作者：
Wannerer, Thomas