  Amenability is a geometric property of convex cones that is stronger than
facial exposedness and assists in the study of error bounds for conic
feasibility problems. In this paper we establish numerous properties of
amenable cones, and investigate the relationships between amenability and other
properties of convex cones, such as niceness and projectional exposure.
  We show that the amenability of a compact slice of a closed convex cone is
equivalent to the amenability of the cone, and prove several results on the
preservation of amenability under intersections and other convex operations. It
then follows that homogeneous, doubly nonnegative and other cones that can be
represented as slices of the cone of positive semidefinite matrices, are
amenable.
  It is known that projectionally exposed cones are amenable and that amenable
cones are nice, however the converse statements have been open questions. We
construct an example of a four-dimensional cone that is nice but not amenable.
We also show that amenable cones are projectionally exposed in dimensions up to
and including four.
  We conclude with a discussion on open problems related to facial structure of
convex sets that we came across in the course of this work, but were not able
to fully resolve.
