ON THE ALEKSANDROV PROBLEM AND THE EXTENSION OF ISOMETRIES BETWEEN UNIT SPHERES

Abstract

We shall prove mappings satisfying WDNPP are not far from being isometries. We also obtain an extension of Mazur-Ulam theorem [26] and Baker theorem [5]. Then, as a special case, the extension of isometries between unit spheres [9] is involved in this kind of problem, we will give two affirmative results on a class of classical Banach spaces based on the main result in Section 2. In the end, mappings preserving equality of distance [3,18,25]without any surjectivity condition are considered.