The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension

Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in ${\mathbb{R}}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d\to \mathrm{\infty }$ as $d\to \mathrm{\infty }$, thus answering, for the case of cubes, a long standing open question of E. M. Stein and J. O. Str"{o}mberg.