Curvature Measures and Integral Geometry

曲率测量和积分几何

基本信息

批准号：
160223306
负责人：
Professor Dr. Daniel Hug
金额：
--
依托单位：
Institut für Stochastik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2010
资助国家：
德国
起止时间：
2009-12-31 至 2015-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/160223306?language=en
关键词：
Curvature Measures Integral Geometry

项目摘要

In the current project, the investigation of curvature measures in euclidean spaces has been advanced in two directions. In a first subproject, curvature related quantities of very general sets have been studied via the approximation with parallel sets and the asymptotic behaviour of such approximations has been explored. Here, very general connections between the asymptotic behaviour of the volume and the surface area of the parallel sets have been found, and this has led to a good understanding of the global aspects of this approximation. Moreover, the existence of fractal curvature measures of self-similar sets has been established in a general framework. In the second subproject, extensions of curvature measures of convex bodies to measures on flag manifolds have been treated. The properties of such flag measures have been studied and it has been discussed how flag measures can be used for classical integral formulas in convex geometry and for the investigation of valuations. Many of the problems that have been addressed are not completely resolved so far. This should be done in the second phase of the project. In particular, we plan to localize, in a measure theoretic setting, the results obtained in the first subproject concerning the connection between volume and surface area of general sets. In addition, we shall study especially those self-similar sets whose curvature measures do not scale with the dimension, with the aim of understanding the geometric meaning of the scaling exponents. In the context of flag measures, further integral representations of mixed volumes and of special functionals (such as the support function) should be developed. Another goal is to find a description of homogeneous valuations by means of flag measures. Finally, applications to stochastic geometry, for instance to Boolean models, still remain to be considered. These applications require the investigation of injectivity properties of operators which are associated with flag measures.

在当前的项目中，对欧几里得空间中曲率度量的研究已在两个方向上进行了提前。在第一个子标记中，已经通过平行集进行了近似研究，研究了非常通用集的曲率相关量，并且已经探索了此类近似值的渐近行为。在这里，已经找到了体积的渐近行为与平行集的表面积之间的非常一般的联系，这使人们对此近似的全局方面有了很好的了解。此外，在一般框架中已经建立了自相似集的分形曲率度量。在第二个子标题中，已经处理了凸体的曲率度量扩展到标志歧管上的测量。已经研究了此类标志测量的特性，并已经讨论了如何将标志度量用于凸几何形状中的经典积分公式和估值的研究。到目前为止，已经解决的许多问题尚未完全解决。这应该在项目的第二阶段完成。特别是，我们计划在一个度量理论环境中定位，在第一个子标记中获得的结果涉及一般集合的音量和表面积之间的连接。此外，我们将特别研究那些曲率度量并非尺寸扩展的自相似集合，以理解缩放指数的几何含义。在FLAG度量的背景下，应开发混合体积和特殊功能（例如支持功能）的进一步积分表示。另一个目标是通过标志措施找到对均匀估值的描述。最后，例如，用于布尔模型的随机几何形状的应用仍然有待考虑。这些应用需要调查与标志测量相关的操作员的注射性能。