Theory And Algorithm Of Estimsting Covering Functionals Of Convex Bodies

Hadwiger proposed the famous Hadwiger’s covering conjecture in 1957 and the conjecture was completely solved only in two dimensions.In order to overcome this conjecture,Professor Zong Chuanming proposed an important program according to the properties of covering functional and its relationship with the covering number of convex bodies.Estimating covering functionals of convex bodies plays an important role in this process.In this paper,we will use theoretical proof and numerical method to estimate covering functionals of convex bodies.In Chapter 2,we obtain the exact values or estimations of covering functionals of the n-dimensional simplices in some cases by using the geometric properties of simplices and the negative homothetic copies covering problem.In Chapter 3,we transform the problem of estimating covering functionals of convex bodies into a vertex p-center problem.By improving the relaxation algorithm proposed by D.Chen et al.,we obtain an exact iterative algorithm to solve the vertex p-center problem.At the same time,some examples are given about the Euclidean unit disk,simplices and the regular octahedron to verify the effectiveness of the algorithm.