Kinematic integral formulas for convex bodies
Kinematic integral formulas for convex bodies
复制标题
凸体运动学积分公式
DOI：
--
复制
发表时间：
1979
期刊：
影响因子：
--
通讯作者：
W. Weil
中科院分区：
文献类型：
--
作者：
W. Weil
下面我们描述积分几何的一些最新发展。凸体的经典积分几何公式以及这些公式的各种推广，读者可以查阅 Hadwiger [1955]、[1957] 和 Santalo [1976] 的书籍，处理相交凸图形。我们的目的是在积分几何的两个最新分支中呈现不同类型的结果。在由 Hadwiger 发起的第一个案例中，人们研究了凸图形的平均值公式，与经典案例相反，凸图形具有正距离。在另一种情况下，可以追溯到 Firey 的工作，人们考虑对凸形图形的接触位置进行测量。这两个主题密切相关。正如我们将看到的，对尽可能通用的第一类型积分公式的搜索会立即导致凸体接触度量的自然定义。此外，由于积分公式以及接触测量都涉及曲率测量，我们的考虑也产生了积分几何的第三个分支的结果，该分支与 Federer [1959]、Schneider [1975]、[1978a] 获得的经典公式的局部版本有关。
In the following we describe some recent developments in integral geometry. The classical integral geometric formulas for convex bodies and the various generalizations of these formulas, for which the reader may consult the books of Hadwiger [1955], [1957] and Santalo [1976], deal with intersecting convex figures. Our aim here is to present results of a different type in two recent branches of integral geometry. In the first case, which was initiated by Hadwiger, one investigates mean value formulas for convex figures which, in contrast to the classical case, have a positive distance. In the other case, which goes back to work of Firey, one considers measures over contact positions of convex figures. Both topics are closely related. As we shall see, the search for integral formulas of the first type that are as general as possible leads one immediately to a natural definition of contact measures of convex bodies. Moreover, since the integral formulas as well as the contact measures involve curvature measures, our considerations also yield results in a third branch of integral geometry, which is concerned with local versions of the classical formulas as they have been obtained by Federer [1959], Schneider [1975], [1978a].