  Given a convex body $K \subseteq \mathbb R^n$ in L\"owner position we study
the problem of constructing a non-negative centered isotropic measure supported
in the contact points, whose existence is guaranteed by John's Theorem. The
method we propose requires the minimization of a convex function defined in an
$\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation
of the minimizer as $\left. \frac{\partial}{\partial r}(A_r,
v_r)\right|_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions
of $K$ that are in some sense related to the maximal intersection position of
radius $r$ defined recently by Artstein-Avidan and Katzin.
