The numerical index of $2$-dimensional Lipschitz-free spaces

Abstract: We provide the explicit formula for the numerical index of any $2$-dimensional Lipschitz-free space, also giving the construction of operators attaining this value as its numerical radius. As a consequence, the numerical index of $2$-dimensional Lipschitz-free spaces can take any value of the interval $[\frac{1}{2},1]$, and this whole range of numerical indices can be attained by taking $2$-dimensional subspaces of any Lipschitz-free space of the form $\mathcal{F}(A)$, where $A\subset {\mathbb{R}}^n$ with $n\geq 2$ is any set with non-empty interior.