We study a family of polytopes and their duals, that appear in various
optimization problems as the unit balls for certain norms. These two families
interpolate between the hypercube, the unit ball for the $\infty$-norm, and its
dual cross-polytope, the unit ball for the $1$-norm. We give combinatorial and
geometric properties of both families of polytopes such as their $f$-vector,
their volume, and the volume of their boundary.