Isometric Decompositions

The study of decompositions of sets in U, n ^ 1, into disjoint, mutually isometric subsets has a long and distinguished history. In 1928 J. von Neumann [2] proved that an interval in U (with or without endpoints) is decomposable into Ko disjoint sets which are mutually isometric under translations. In 1951 W. Gustin [1] proved that no such decomposition into n, 2 ^ n < No, sets is possible. A brief summary of known results in the area is contained in a recent paper [3] by S. Wagon. Also in [3] it is shown that a ball in U, n ^ 2, is not decomposable into m sets, where 2 ^ m ^ n, which are mutually isometric under rotations and translations of U. In this paper we treat some related problems concerning decompositions of a closed unit ball in Banach space. To facilitate the discussion we say that a set S (in a metric space) is m-decomposable if S is the union of m mutually disjoint subsets and for any two of them there is a distance preserving mapping from one onto the other. Where appropriate a statement about the type of mappings is included; for example 'm-decomposable under translations', etc. We deal briefly with metric spaces which admit an isometry satisfying a special condition (cf. Section 2). Such spaces are shown to be m-decomposable for any cardinal m, 2 ^ m ^ Ko. Our main interest, however, lies in settling some of the numerous and difficult questions pertaining to the isometric decompositions of the closed unit ball in certain Banach spaces. In strictly convex Banach spaces X the closed unit ball is shown not to be m-decomposable, 2 ^ m < Xo, under translations. If X is an inner-product space then, in addition, the closed unit ball is not K0-decomposable. Further, under the family of all isometries, the closed unit ball of a finitedimensional strictly-convex Banach space X is shown not to be m-decomposable for any m with 2 < m ^ n, where n = dim X. In contrast, in such non-strictly-convex Banach spaces as c0 and C[0,1] we show that the closed unit ball is indeed m-decomposable under certain isometries for all m with 2 ^ m ^ Ko.