  For plane strain linear elasticity, given any anisotropic elasticity tensor
$\mathbb{C}_{\rm aniso}$, we determine a best approximating isotropic
counterpart $\mathbb{C}_{\rm iso}$. This is not done by using a distance
measure on the space of positive definite elasticity tensors (Euclidean or
logarithmic distance) but by considering two simple isotropic analytic
solutions (center of dilatation and concentrated couple) and best fitting these
radial solutions to the numerical anisotropic solution based on
$\mathbb{C}_{\rm aniso}$. The numerical solution is done via a finite element
calculation, and the fitting via a subsequent quadratic error minimization.
Thus, we obtain the two Lam\'e-moduli $\mu$, $\lambda$ (or $\mu$ and the
bulk-modulus $\kappa$) of $\mathbb{C}_{\rm aniso}$. We observe that our
so-determined isotropic tensor $\mathbb{C}_{\rm iso}$ coincides with neither
the best logarithmic fit of Norris nor the best Euclidean fit. Our result calls
into question the very notion of a best-fit isotropic elasticity tensor to a
given anisotropic material.
