  We introduce a class of convex, higher-dimensional billiard models which
generalise stadium billiards. These models correspond to the free motion of a
point-particle in a region bounded by cylinders cut by planes. They are
motivated by models of particles interacting via a string-type mechanism, and
confined by hard walls. The combination of these elements may give rise to a
defocusing mechanism, similar to that in two dimensions, which allows large
chaotic regions in phase space. The remaining part of phase space is associated
with marginally stable behaviour. In fact periodic orbits in these systems
generically come in continuous parametric families, sociated with a pair of
parabolic eigen-directions: the periodic orbits are unstable in the presence of
a defocusing mechanism, but marginally stable otherwise. By performing the
stability analysis of families of periodic orbits at a nonlinear level, we
establish the conditions under which families are nonlinearly stable or
unstable. As a result, we identify regions in the parameter space of the models
which admit non-linearly stable oscillations in the form of whispering gallery
modes. Where no families of periodic orbits are stable, the billiards are
completely chaotic, i.e.\ the Lyapunov exponents of the billiard map are
non-zero.
