A lower bound for the equilateral number of normed spaces

We show that if the Banach-Mazur distance between an $n$-dimensional normed space $X$ and $\ell_\infty^n$ is at most $3/2$, then there exist $n+1$ equidistant points in $X$. By a well-known result of Alon and Milman, this implies that an arbitrary $n$-dimensional normed space admits at least $e^{c\sqrt{\log n}}$ equidistant points, where $c>0$ is an absolute constant. We also show that there exist $n$ equidistant points in spaces sufficiently close to $\ell_p^n$, $1<p<\infty$.