The Aleksandrov Problem About Preserving Distance Mapping

In 1970, A.D.Aleksandrov posed the following question:whether the existence of a single conservative distance for some f implies that f is an isometry from a metric space E into itself? It is called the Aleksandrov problem.The Aleksandrov problem has been extensively studied by many mathematical workers and some beatiful conclusions have been obtained. But, until now, there are still a lot of problems unsolved. In this thesis, we study the Aleksandrov problem with analytic method in the quasi-convex normed linear spaces, Hilbert space and Rn space and obtain some important results and answer this problem partially. We have divided this thesis into four chapters. In the first chapter, we briefly introduce the research background, current situation on the Aleksandrov problem, and the main result of this paper. In the second chapter, we introduce preliminary knowledge about the quasi-convex normed space, Hilbert space and Rn space and the properties of mapping f The third chapter is devoted to the Aleksandrov problem in different spaces. This chapter is divided into three sections, in the first section, we mainly research on the Aleksandrov problem in the quasi convex normed space, we proved that:(1) Let X and Y be two quasi convex normed linear spaces, if f:X → Y pre-serves collinearity and satisfies that (i)||x-y||≤ 1, then||f(x)-f(y)||≤||x-y||; (ii)|| x-y||> no, then||f(x)-f(y)||> no. then f is an affine isometry.(2) Let X and Y be two quasi convex 2-normed linear spaces, if f:X → Y satisfies(GAOPP)and ||f(x)-f(z),f(P)-f(q)||≤||x-z,p-q|| for all x,z,p,q ∈ Xwith||x-z,p-q||≤ 1, then f is an affine generalized 2-isometry.(3)Let X and Y be two quasi convex 2-normed linear spaces, and f:X ∈→ Y, if f has GAOPP, it is positively homogeneous and f preserves collinearity, then it is a 2-isometry.(4)Let X and Y be two quasi convex 2-normed linear spaces. Suppose that f X → Y be affine and it preserve all areas m< 1. Then f is an 2-isometry.The second section, is devoted to the Aleksandrov problem in Hibert space, the main results:(1)Let X and Y be Hibert spaces, dimX≥ 2, assume that f:X → Y preserves two distances 1 and then f is a linear isometry up to translation.(2)LetX and Y be Hibert spaces, dimX≥ 2, assume that f:X → Y preserves two distancesl and hen f is a linear isometry up to translation.In the third section, we considered the Aleksandrov problem in Rn space. we proved that:Assume that f:Rn-1 → Rn preserves two distances 1 andthen f is a linear isometry up to translation.