This is an announcement for the paper "Equilateral dimension of some classes of normed spaces" by Tomasz Kobos.
Abstract: An equilateral dimension of a normed space is a maximal number of pairwise equidistant points of this space. The aim of this paper is to study the equilateral dimension of certain classes of finite dimensional normed spaces. The well-known conjecture states that the equilateral dimension of any $n$-dimensional normed space is not less than $n+1$. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces, Orlicz-Musielak spaces satisfying an additional condition and every $(n-1)$-dimensional subspace of $\ell^{n}_{\infty}$ we also provide some weaker bounds on the equilateral dimension for every space which is sufficiently close to one of these. This generalizes the result of Swanepoel and Villa concerning the $\ell_p^n$ spaces.
Archive classification: math.MG math.FA
Mathematics Subject Classification: 46B85, 46B20, 52C17, 52A15, 52A20
Remarks: 12 pages
Submitted from: tkobos@wp.pl
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1305.6288
or
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alspach＠math.okstate.edu