  Let M be a polynomially bounded, o-minimal structure with archimedean prime
model, for example if M is a real closed field. Let C be a convex and unbounded
subset of M. We determine the first order theory of the structure M expanded by
the set C. We do this also over any given set of parameters from M, which
yields a description of all subsets of M^n, definable in the expanded
structure.
