On the (m, n)-clock problem and the \(\ell _{\infty }-\ell _1\) norm of a matrix

We characterize the norm attainment set of a linear operator from \( \ell _{\infty }^{2}({\mathbb {C}}) \) to \( \ell _{1}^{2}({\mathbb {C}}), \) with the help of a physical model involving two clocks entangled in a specific way. More generally, we introduce the (m, n)-clock Problem and establish its equivalence with computing the \(\ell _{\infty }-\ell _1\) norm of an \( m \times n \) matrix. We further give an explicit description of the smooth and the non-smooth points in \({\mathbb {L}}\big (\ell _\infty ^2({\mathbb {C}}),\ell _1^2({\mathbb {C}})\big ).\)