A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of Mk, definable in the expansion ℳ of M by all convex subsets of the line. We show that ℳ after naming constants, is model complete provided M is model complete.
Marcus Tressl. "Heirs of box types in polynomially bounded structures." J. Symbolic Logic 74 (4) 1225 - 1263, December 2009. https://doi.org/10.2178/jsl/1254748689