Event Details
- IMAG Functional Analysis Seminar
- Title:Klee's tiling of Gamma. Variations on a theme.
- By Tommaso Russo (Universität Innsbruck, Austria)
- Abstract: In 1981 Victor Klee proved that \ell_1(\Gamma), for |\Gamma|=\mathfrak{c}, can be written as a disjoint union of closed balls of radius 1. In particular, the set of centers of the balls is a discrete Chebyshev set. In the first part of the talk we will explain the main geometric idea behind this construction and also some details of the proof (depending on the audience's interest).Subsequently, we will explain some variants of the construction, that, in a sense, try to reproduce the same construction in the Hilbert space \ell_2(\Gamma). The main output of this construction is that there is a symmetric, bounded, convex body whose translates form a tiling of \ell_2(\Gamma). By this, we mean that the convex bodies cover the space and their interiors are mutually disjoint. In particular, we obtain the existence of a reflexive Banach space (isomorphic to \ell_2(\Gamma)) that admits a tiling by balls of radius 1, which answers a question due to Fonf and Lindenstrauss. Another variant (actually, a simplification) of the construction gives an alternative self-contained proof of a result due to Dilworth, Odell, Schlumprecht, and Zsák that every Banach space admits a net which is a subgroup.
