  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.
