Gelfand width of $l_1$ unit ball in $\infty$ metric

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The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not very tight. Can somebody point me to a source? The main reference on the subject, Carl and Stephani monograph does not help in this case either.