Abstract of a paper by Tomasz Kania and Tomasz Kochanek
This is an announcement for the paper "Uncountable sets of unit vectors that are separated by more than 1" by Tomasz Kania and Tomasz Kochanek. Abstract: Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a compact, Hausdorff space, then $X=C(K)$ contains such a set of cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis. Archive classification: math.FA math.MG Remarks: 17 pp Submitted from: tomasz.marcin.kania@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1503.08166 or http://arXiv.org/abs/1503.08166
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alspach@math.okstate.edu