Properties And Estimations Of Covering Functionals Of Convex Bodies

The Hadwiger’s covering conjecture on convex bodies is an open problem that has lasted for more than 60 years.The conjecture is closely related to two kinds of covering problems and two kinds of boundary illuminations of convex bodies.many scholars have used a series of equivalent forms while trying to prove this conjecture.The substantial progress of Hadwiger’s conjecture is not only essential to the complete solution of this conjecture,but also to the solution of related problems in convex and discrete geometry and the development of correspondence theory.On the basis of the existing research,this thesis makes a further study on the properties and estimation of the covering functional of convex bodiesFirst of all,research status and development trends at home and abroad of Hadwiger’s conjecture and the covering functional of convex bodies are presented,including covering problems of convex bodies,equivalent forms of Hadwiger’s conjecture,some estimates of c(K),the four-step quantitative program proposed by Chuanming Zong to conquer Hadwiger’s conjecture,and the definition,properties and existing estimates of covering functionalsSecondly,for a convex body K and two positive integers m and l satisfying m ≤l,it is shown that the smallest positive number γ1 such that K can be covered can be dominated by m translates of γ1K by the infimum γ2 of all positive numbers such that the set of extreme points of K can be covered by l translates of γ2KAt the same time,a series of inequalities concerning covering functional of convex bodies having the form K+L,where K is a convex body and L is a segment.The estimates of covering functionals of direct sums of convex bodies are also presentedFinally,the research in this thesis is summarized.