The mean random chord length of convex sets and their extreme values(凸域内的平均随机弦长及其极值)

Abstract

In order to analyze the mean random chord length of convex sets under different kinds of random processes, we take circles, equilateral triangles, rectangles, and squares as examples. Their mean values are obtained using definition method. Then the extreme values of the mean chord length of convex sets are discussed by means of the chord power integrals and their inequalities. Furthermore, some inequalities about these mean values are established, and two conjectures are proposed.(为研究在不同随机意义下平面凸域内的平均随机弦长问题，以圆域、正三角形域、矩形域、正方形域为例，用定义法得到这些凸域内的各平均随机弦长。用弦幂积分及其不等式，讨论了任意凸域内5种平均随机弦长的极值问题，并建立了相应的不等式。在此基础上，提出了2个猜想。)

Keywords

• mean random chord length(平均随机弦长)
• extreme value(极值)
• convex set(凸域)
• mean distance(平均距离)
• chord power integral(弦幂积分)
• integral geometry(积分几何)