轴对称凸域的弦协差及其应用

The Convariogram of Axisymmetric Convex Domains and Applications

为了研究轴对称凸域的弦协差,以正五边形域为例,利用广义支持函数得到了正五边形域弦协差的具体解析式,给出了正五边形域的定向弦长分布函数的具体结果.通过建立凸体的弦协差与运动测度之间的关系式,得到了正五边形域中,当小针长度不超过其边长时,小针含于此正五边形域内的运动测度.在此基础上,分别给出无向小针、有向小针含于正五边形域内的几何概率.

In order to analyze the convariograms of axisymmetric convex domains, taking the regular pentagon as an example, the explicit expression of the covariogram for a regular pentagon is obtained by using the generalized support function. The orientation-dependent chord length distribution function of the regular pentagon is given by using the covariogram. Further, through establishment of a relationship between the covariogram and the kinematic measure, a needle within a regular pentagon is given, when the length of needle does not exceed the side length of the regular pentagon. On this basis, the probabilities of a random segment, both undirected and directed, and entirely lying in a regular pentagon are given respectively.