The Numerical Index of Two-Dimensional Real ${\mathrm{\ell }}_p$ Spaces

The computation of the numerical index of classical Banach spaces is one of the original problems in the theory. In this paper, we compute the numerical index of two-dimensional real ${\mathrm{\ell }}_p$-spaces for all $p\ge 1$. More precisely, we prove that $$n({\mathrm{\ell }}_p^2)=v(J),J=\left(\begin{array}{ccc}0& 1-1& 0\end{array}\right),$$ confirming the conjectured formula in the two-dimensional real case.