Given a lattice $L$ in a Banach space $(B,\|\;\|)$, one denotes by $\lambda_1(L)$ the least norm of a nonzero element in $L$, and by $\lambda_k$ the least $\lambda$ such that there is a linearly independent set of $k$ elements in $L$ each of which has norm at most $\lambda$. Denote by $L^\ast$ the dual lattice in the dual Banach space, equipped with the norm $\|\;\|^\ast$. Let $b=\dim B$. The product $\lambda_1(L,\|\;\|)\,\lambda_b(L^\ast,\|\;\|^\ast)$ is scale-invariant. For Euclidean lattices, it is not hard to show that the maximal value of $\lambda_1(L)\,\lambda_2(L^\ast)$ for $b=2$ is $\sqrt{\frac43}$ (the Hermite constant in this dimension). What is the maximal value for general lattices in 2-dimensional Banach space? Using the John ellipsoid, one can get an upper bound of $\sqrt{\frac83}$, but is it optimal?