Polar reciprocal convex bodies

The minimum of the product of the volume of a symmetric convex bodyK and the volume of the polar reciprocal body ofK relative to the center of symmetry is attained for the cube and then-dimensional crossbody. As a consequence, there is a sharp upper bound in Mahler’s theorem on successive minima in the geometry of numbers. The difficulties involved in the determination of the minimum for unsymmetricK are discussed. Reserch partially supported by NSF Grant GP-27960. An erratum to this article is available at .