Estimations Of Covering Functionals Of Two Classes Of Convex Bodies

In 1957,Hadwiger posed a conjecture.This conjecture has been studies by scientists such as I.Gohberg and A.Markus and so all.Previous work shows that the inequality part of Hadwiger conjecture is true if and only if the smallest positive number ? such that convex body K in n? can be covered by 2n translates of ?K is less than 1.Based on this conclusion,this paper studies the cover functional of the unit circle XS of a Minkowski plane X from theory and the cover functional of the convex cone from algorithm.Firstly,We amend a widely used result given by Doyle,Lagarias,and Randall concerning the side length of equilateral Minkowski m-gons inscribed in the unit circle of a Minkowski plane.Based on this,we obtain the smallest positive number ? such that the unit circle XS of a Minkowski plane X can be covered by m translates of X?B,where XB is the unit ball of X.Moreover,we improve a recent estimation of the smallest positive number ? such that the unit ball XB of a Minkowski space X can be covered by m translates of X?B.Secondly,in this paper,the linear programming algorithm is used to solve the cover functional problem of the convex cone.Estimated values of()m? ?,m =5,6,(43),16 of 4-dimensional convex cones whose bases are the regular tetrahedron,the unit ball of 1l norm and the unit ball of 2l norm in 3-dimensional space are given.Experimental results showed that the method of this article to solve the value of covering functional is best possible for some special convex cones.