We prove some rather precise renorming theorems for Banach spaces with Szlenk index ω0. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.

Let M be a triangulable compact manifold. We prove that, among closed subgroups of Homeo0(M) (the identity component of the group of homeomorphisms of M), the subgroup consisting of volume preserving elements is maximal. AMS classification. 57S05 (57M60, 37E30).

I 1924, Banach and Tarski (1) accomplished a rather paradoxical feat. They proved that a solid ball can be decomposed into five pieces, which are then moved around and reassembled in such a way as to obtain two balls identical to the original one (1). This wellnigh miraculous duplication was based on Hausdorff’s (2) 1914 work. In his 1929 study of the Hausdorff–Banach–Tarski paradox, von Neuman...