凸体的覆盖与照亮

Covering and illumination of convex bodies

关于凸体覆盖的Hadwiger猜想是源于凸和离散几何的悬而未决的公开问题。在综述有关该猜想的经典结果之后,介绍用以研究这一猜想的两种不同方案及近期的相关结果。第一种是宗传明提出的数量化方案,它的可行性等价于Hadwiger猜想的正确性。另一方案基于对高维中心对称凸体的覆盖问题的研究,其可行性取决于P.Soltan的一个问题是否有肯定的回答。

After a survey of classical results on Hadwiger’s covering conjecture,a long-standing open problem from convex and discrete geometry,we introduced two approaches to attack this conjecture and their related results.The first approach is Chuanming Zong’s quantitative program,which is theoretically feasible if Hadwiger’s covering conjecture is true.The other tries to confirm this conjecture by attacking it for high-dimensional centrally symmetric convex bodies,whose feasibility depends on the affirmative answer to a problem posedby P.Soltan.