  Mahler's conjecture asks whether the cube is a minimizer for the volume
product of a body and its polar in the class of symmetric convex bodies in a
fixed dimension. It is known that every Hanner polytope has the same volume
product as the cube or the cross-polytope. In this paper we prove that every
Hanner polytope is a strict local minimizer for the volume product in the class
of symmetric convex bodies endowed with the Banach-Mazur distance.
