Estimations Of Covering Functionals Of A Class Of Convex Bodies

In 1957,Hadwiger proposed the famous Hadwiger's covering conjecture.The conjecture consists of two parts:first,for any convex body K?Kn,the minimum number c(K)required to cover K by translates of int K is bounded from the above by 2n;second,c(K)=2n if and only if K is a parallelotope.After the conjecture was put forward,many scholars have studied it and obtained some results.However,it is very difficult to solve this conjecture because c(K)is discontinuous,only upper semicontinuous.Later,Professor Zong Chuanming introduced the covering functional ?m(K),which is the smallest positive real number ? such that K can be covered by m translates of yK,where m is a positive integer.Since c(K)?2n if and only if ?2n(K)<1,the Hadwiger's covering conjecture can be studied via estimating ?m(K).In the first part,we review the origin of Hadwiger's conjecture,give some equivalent forms of Hadwiger's conjecture and upper bound estimates of c(K)for some special convex bodies,and present some related research and recent developments of covering functionals.The second part gives some knowledge and notation from convex geometry.In the third part,we continue Wu Senlin and Zhou Ying's work to study a class of convex bodies,which is the convex hull of two or more compact convex sets.By studying the covering functionals and the covering of the set of extreme points of each compact convex sets,we give estimations of the covering functionals of this class of convex bodies.It is proved that when a three-dimensional convex body K is the convex hull of two compact convex sets having no interior points,c(K)is not greater than 8.In this class of convex bodies,c(K)=8 if and only if K is a parallelepiped.