Curvature Measures and Integral Geometry

曲率测量和积分几何

• 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

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，欧几里得空间中曲率测度的研究已在两个方向上推进。在第一个子项目中，通过平行集合的近似研究了非常一般集合的曲率相关量，并探索了这种近似的渐近行为。在这里，我们发现了体积的渐近行为和平行集的表面积之间非常普遍的联系，这使得我们能够更好地理解这种近似的全局方面。 此外，自相似集分形曲率测度的存在性已在一般框架中得到确立。在第二个子项目中，处理了凸体曲率测量到旗形流形测量的扩展。已经研究了此类标志测度的属性，并讨论了如何将标志测度用于凸几何中的经典积分公式以及评估研究。迄今为止，许多已经解决的问题还没有完全解决。这应该在项目的第二阶段完成。特别是，我们计划在测量理论环境中本地化第一个子项目中获得的关于一般集合的体积和表面积之间的联系的结果。此外，我们将特别研究那些曲率度量不随维数缩放的自相似集，目的是理解缩放指数的几何意义。在标志测量的背景下，应该开发混合体积和特殊函数（例如支持函数）的进一步积分表示。另一个目标是通过标志度量找到同质估值的描述。最后，随机几何的应用，例如布尔模型，仍有待考虑。这些应用需要研究与标志测量相关的算子的注入性属性。