  Pseudo-cones are a class of unbounded closed convex sets, not containing the
origin. They admit a kind of polarity, called copolarity. With this, they can
be considered as a counterpart to convex bodies containing the origin in the
interior. The purpose of the following is to study this analogy in greater
detail. We supplement the investigation of copolarity, considering, for
example, conjugate faces. Then we deal with the question suggested by
Minkowski's theorem, asking which measures are surface area measures of
pseudo-cones with given recession cone. We provide a sufficient condition for
possibly infinite measures and a special class of pseudo-cones.
