  In this paper, we study properties of certain risk measures associated with
acceptance sets. These sets describe regulatory preconditions that have to be
fulfilled by financial institutions to pass a given acceptance test. If the
financial position of an institution is not acceptable, the decision maker has
to raise new capital and invest it into a basket of so called eligible assets
to change the current position such that the resulting one corresponds with an
element of the acceptance set. Risk measures have been widely studied. The risk
measure that is considered here determines the minimal costs of making a
financial position acceptable. In the literature, monetary risk measures are
often defined as translation invariant functions and, thus, there is an
equivalent formulation as Gerstewitz-Functional. The Gerstewitz-Functional is
an useful tool for separation and scalarization in multiobjective optimization
in the non-convex case. In our paper, we study properties of the sublevel sets,
strict sublevel sets and level lines of a risk measure defined on a linear
space. Furthermore, we discuss the finiteness of the risk measure and relax the
closedness assumptions.
