The Best Constant For The Besicovitch Covering Theorem And Kissing Number

In studying many analysis problems, various covering theorems have played a very important role.In forties of 20th century,Besicovitch first demonstrated his result for disks in the plane.This is the original theorem of Besicovitch.The next two years, Morse ex-tended it to the more general spaces, which expanded the application range of the theorem undoubtedly. And in the process of applying the theorem, we found that the two constants in the theorem is a problem that is worth to study. The famous Kissing number problem asks for the maximal value of the same scale nonoverlapping balls in n-dimensional space that can kiss another ball of the equal scale. Early in nineties of 17th century,Isaac New-ton and David Gregory once discussed the condition in 3-dimensional space.In this paper, we analyse the best constant for the Besicovitch covering theorem and the Kissing number in low-dimensional space, then we observe that there is a certain special relationship be-tween them in corresponding space, thus we guess that the similar relationship also exists between them in high-dimensional space. finally we give a relevant proof. In other words, we use the existing results of the Kissing number so that we can give an appropriate value or bound to the best constant for the Besicovitch covering theorem, which has far-reaching significance to the application of the theorem.