On the norming constants for normal maxima

In a remarkable paper, Peter Hall [{\it On the rate of convergence of normal extremes}, J. App. Prob, {\bf 16} (1979) 433--439] proved that the supremum norm distance between the distribution function of the normalized maximum of $n$ independent standard normal random variables and the distribution function of the Gumbel law is bounded by $3/\log n$. In the present paper we prove that choosing a different set of norming constants that bound can be reduced to $1/\log n$. As a consequence, using the asymptotic expansion of a Lambert $W$ type function, we propose new explicit constants for the maxima of normal random variables.