It is proved that the simplex is a strict local minimum for the
volume-product P(K)=min vol(K)vol(K^z), in the Banach-Mazur space of
n-dimensional (classes of) convex bodies. Here K^z is the polar body of K about
the point z and the minimum is taken over all the points z in the interior of
K. Linear local stability in the neighborhood of the simplex is proved as well.
In the proof, methods that were recently introduced by Nazarov, Petrov,
Ryabogin and Zvavitch are extended to the non-symmetric setting.