  A translation body of a convex body is the convex hull of two of its
translates intersecting each other. In the 1950s, Rogers and Shephard found the
extremal values, over the family of $n$-dimensional convex bodies, of the
maximal volume of the translation bodies of a given convex body. In our paper,
we introduce a normed version of this problem, and for the planar case,
determine the corresponding quantities, with one exception, for the four types
of volumes regularly used in the literature: Busemann, Holmes-Thompson, and
Gromov's mass and mass*. We examine the problem also for higher dimensions, and
for centrally symmetric convex bodies.
