  Domain walls and droplet-like excitation of the random-field Ising magnet are
studied in d={3,4,5,6,7} dimensions by means of exact numerical ground-state
calculations. They are obtained using the established mapping to the
graph-theoretical maximum-flow problem. This allows to study large system sizes
of more than five million spins in exact thermal equilibrium. All simulations
are carried out at the critical point for the strength h of the random fields,
h=h_c(d), respectively. Using finite-size scaling, energetic and geometric
properties like stiffness exponents and fractal dimensions are calculated.
Using these results, we test (hyper) scaling relations, which seem to be
fulfilled below the upper critical dimension d_u=6. Also, for d<d_u, the
stiffness exponent can be obtained from the scaling of the ground-state energy.
