Weak minimizing property on pairs of classical Banach spaces

We investigate the minimum modulus analogue of the weak maximizing property, termed the \emph{weak minimizing property}. We establish that the pairs $(\ell_p, L^p[0, 1])$ for $2 \leq p < \infty$ and $(\ell_s \oplus_q \ell_q, \ell_r \oplus_p \ell_p)$ for $1 < p \leq r\leq s \leq q < \infty$ satisfy the weak minimizing property. Conversely, we prove that the pairs $(\ell_1, \ell_p)$, $(\ell_1, c_0)$, $(\ell_1, \ell_1)$ and $(c_0, \ell_p)$ fail to satisfy the weak minimizing property.