The intersection body of a ball is again a ball. So, the unit ball $B_d
\subset \R^d$ is a fixed point of the intersection body operator acting on the
space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur
distance.We show that this fixed point is a local attractor, i.e., that the
iterations of the intersection body operator applied to any star-shaped origin
symmetric body sufficiently close to $B_d$ in Banach-Mazur distance converge to
$B_d$ in Banach-Mazur distance.