Problems Related To Illuminated Caps Of The Unit Sphere

In the mid-twentieth century, Hadwiger posed the famous Hadwiger's covering conjecture(H-conjecture for short). In recent years, although many mathematicians have done a lot of important work concerning H-conjecture, this conjecture only was solved only in two-dimensional spaces. There is no doubt that solving the conjecture still need new ideas and new methods. Therefore, this thesis will mainly study problems related to illuminated caps of the unit sphere, and study Hadwiger's conjecture for centrally symmetric convex bodies.First, this thesis reviews the origin of Hadwiger conjecture, outlines existing results concerning H-conjecture, survey various methods that estimate the covering number c(K) by studying functionals definded on the space of convex bodies, and introduce known results concerning the generalized blocking number.Second, in view of the important role that illuminated caps plays in the study of Hadwiger's covering problem for centrally symmetric convex bodies, basic properties of illuminated caps of the unit sphere are studied, their geometric shapes are characterized. Moreover, related characterizations of inner product spaces are studied and two characterizations of inner product spaces among strictly convex normed linear spaces are presented. The maximality of illuminated caps is studied, a necessary and sufficient condition for an illuminated cap to be maximal is obtained.It is also shown that each illuminated cap of a smooth normed linear space is maximal.The final part presents definitions of incenter and inner radius, clarifies the existence of incenter. The circumradius of an illuminated cap is defined, the bounds of the circumradius are presented and characterizations of the situation when the lower and upper bound are attained are obtained. Two new geometric constants are introduced via maximality of illuminated caps and the inner radius of them. The bounds of these two constants and the relation between them are clarified.