Abstract

Let $T$ be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of $T$ and show that the residue field of such a convex hull has a natural expansion to a model of $T$. We give a quantifier elimination relative to $T$ for the theory of pairs $$ where $\mathscr{R} \models T$ and $V \neq \mathscr{R}$ is the convex hull of an elementary substructure of $\mathscr{R}$. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to $T$ for the theory of pairs $$ with $\mathscr{R}$ a model of $T$ and $\mathscr{N}$ a proper elementary substructure that is Dedekind complete in $\mathscr{R}$. We deduce that the theory of such "tame" pairs is complete.