Covering Functionals Of Several Classes Of Special Convex Bodies

Since the Hadwiger conjecture was put forward,it has become a famous difficult problem in the field of convex and discrete geometry.Because the smallest positive number?such that convex body K can be covered by m translates of?K is less than 1 in n-dimensional Euclidean space is equivalent to the covering number of K does not exceed m.It is an important work to estimate the covering functionals of convex bodies.In view of this,we have completed the estimation of the covering functionals of convex cones and double cones.Firstly,this paper reviews the origin of Hadwiger's conjecture,summarizes the progress of previous studies in solving this open problem,and introduces some important conclusions related to it.Secondly,breaking through the limitation of space dimension,the relationship between covering functionals of convex cones and double cones and the covering functionals of the bottom is given.The inequality relation between covering functionals?38?_m?10?1?7?C?8?and?38?_m?7?K?8?of convex cones with K as base is established.The three-dimensional result is better than the corresponding result of Chuanming Zong's.Because estimating the covering functionals of convex bodies in n-dimensional Euclidean space is very complex,some special three-dimensional convex cones are calculated and the exact values of convex cones with parallelogram as the base are given when m?28?5.Finally,The inequality relation between covering functionals)83(_m?10?2?7?C?8?and)83(_m?7?K?8?of double cones with K as base is established.By comparing with the exact values of some special double cones obtained by predecessors,the estimates given in this paper are optimal in some cases.