  For a natural number d and a d-dimensional real vector r let Tau(r) denote
the (d-dimensional) shift radix system associated with r. Tau(r) is said to
have the finiteness property iff all orbits of Tau(r) end up in the zero
vector; the set of all corresponding parameters r is denoted by DN(d), whereas
D(d) consists of those parameters r for which all orbits are eventually
periodic. DN(d) has a very complicated structure even for d=2. In the present
paper two algorithms are presented which allow the characterization of the
intersection of DN(d) and any closed convex hull of finitely many interior
points of D(d) which is completely contained in the interior of D(d). One of
the algorithms is used to determine the structure of DN(2) in a region
considerably larger than previously possible, and to settle two questions on
its topology: It is shown that DN(2) is disconnected and that the largest
connected component has nontrivial fundamental group. The other algorithm is
the first characterizing DN(d) in a given convex polyhedron which terminates
for all inputs. Furthermore several infinite families of "cutout polygons" are
deduced settling the finiteness property for a chain of regions touching the
boundary of D(2).
