Generalizations of the centroid with an application in stochastic geometry
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- 作者：Daniel Vašata
- 刊名：Mathematische Nachrichten
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：290
- 期：2-3
- 页码：452-473
- 全文大小：375K
- ISSN：1522-2616
文摘
The centroid of a subset of with positive volume is a well-known characteristic. An interesting task is to generalize its definition to at least some sets of zero volume. In the presented paper we propose two possible ways how to do that. The first is based on the Hausdorff measure of an appropriate dimension. The second is given by the limit of centroids of ε-neighbourhoods of the particular set when ε goes to 0. For both generalizations we discuss their existence and basic properties. Then we focus on sufficient conditions of existence of the second generalization and on conditions when both generalizations coincide. It turns out that they can be formulated with the help of the Minkowski content, rectifiability, and self-similarity. Since the centroid is often used in stochastic geometry as a centre function for certain particle processes, we present properties that are needed for both generalizations to be valid centre functions. Finally, we also show their continuity on compact convex m-sets with respect to the Hausdorff metric topology.