  In this paper we consider the problem of minimizing the relative perimeter
under a volume constraint in the interior of a conically bounded convex set,
i.e., an unbounded convex body admitting an \emph{exterior} asymptotic cone.
Results concerning existence of isoperimetric regions, the behavior of the
isoperimetric profile for large volumes, and a characterization of
isoperimetric regions of large volume in conically bounded convex sets of
revolution is obtained.
