Approximation Of Convex Bodies In 2,3 Dimensions

The main content of this paper is to estimate the Banach-Mazur distance between regular polygons and the simplex. The estimation of the best upperbound for the Banach-Mazur distance between convex bodies is a long-standing problem, which has attracted much attention of mathematicians for many years. Many kinds of approaches to challenging this problem have been created, among which the popular one is to estimate the distance between two particular classes of convex bodies first and then to estimate it for general convex bodies.In this thesis, we try to use a new approach to estimating the Banach-Mazur distance, concretely, we prove first that for any compact setsC,D(?) R~n及λ>1,∩y∈C D(λ,y)≠Φif and only if for some x∈Rn, where C-x(?)λ(D-x),其中D(λ,x)=λ/λ-1(D-x)+x With such a fact, we provide a simple proof for Lassak's estimate for the Banach-Mazur distance between regular pentagons and the simplex. By the same method, the Banach-Mazur distance between regular heptagon and the simplex is also obtained.The most important advantage of the approach used in this thesis is that it can be actually used to get the Banach-Mazur distances between regular odd-sides polygons and the simplex, and provides at the same time a possible method to estimate the distance between general convex bodies and the simplex.