A structure M is o-minimal if and only if every formula with one free variable and parameters in M is equivalent to a quantifier-free formula involving only the ordering, also with parameters in M. This is analogous to the minimal structures, which are exactly the analogous property down to equality.
Thus the study of o-minimal structures and theories generalises real algebraic geometry.
A major line of current research is based on discovering expansions of the real ordered field that are o-minimal.
Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures.
Moreover, continuously differentiable definable functions in a o-minimal structure satisfy a generalization of Łojasiewicz inequality,[7] a property that has been used to guarantee the convergence of some non-smooth optimization methods, such as the stochastic subgradient method (under some mild assumptions).