  Bounded interactions are particularly important in soft-matter systems, such
as colloids, microemulsions, and polymers. We derive new duality relations for
a class of soft potentials, including three-body and higher-order functions,
that can be applied to ordered and disordered classical ground states. These
duality relations link the energy of configurations associated with a
real-space potential to the corresponding energy of the dual
(Fourier-transformed) potential. We apply the duality relations by
demonstrating how information about the classical ground states of short-ranged
potentials can be used to draw new conclusions about the ground states of
long-ranged potentials and vice versa. The duality relations also lead to
bounds on the T=0 system energies in density intervals of phase coexistence.
Additionally, we identify classes of "self-similar" potentials, for which one
can relate low- and high-density ground-state energies. We analyze the ground
state configurations and thermodynamic properties of a one-dimensional system
previously thought to exhibit an infinite number of structural phase
transitions and comment on the known ground states of purely repulsive
monotonic potentials in the context of our duality relations.
