Problems Related To Covering And Illumination Of Convex Bodies

In1957Hadwiger posed Hadwiger’s covering conjecture (H-conjecture forshort). Although many mathematicians have done a lot of important work concerningH-conjecture, and this conjecture has also been repeatedly mentioned in many surveypapers and monographs, H-conjecture is still not very well understood. Thisconjecture is essentially open even for the three-dimensional case and we are clearlyfar away from the complete solution of it. There is no doubt that solving theconjecture still need more new ideas as well as a long-term hard work. In view of thissituation, this thesis will mainly study problems related to covering and illuminationof convex bodies, and try to contribute to the solution of H-conjecture.Firstly, this thesis briefly reviews the origin of H-conjecture, its equivalentforms, some attempts to attack H-conjecture via direct estimates of the minimumnumber c (K)of translates of the interior of a convex body K needed to cover K.These basic results laid a solid foundation for the discussion in the sequel.The first part of the main results uses two different methods to prove that c(K)equal to the minimum number of smaller homothetic copies of a convex body Kneeded to cover the boundary of K, which is firstly announced by M. Lassak in1988without a detailed proof.As the second part of the main result, we prove that when the boundary of theconvex body can be illuminated by several directions (which can be seen as anumber of parallel light beam). The “width” of these light beams can be uniformlycompressed. Based on this, a new proof of the fact that the minimum number ofsmaller homothetic copies of a convex body K needed to cover K equals to theminimum number of directions needed to illuminate the boundary of K.Finally, we give an estimate of the upper bound of c (K)for a special class ofthree-dimensional convex bodies.