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The Expressivity of Quantifying over Regions

We categorize in recursion-theoretic terms the expressivity of a number of first-order languages that allow quantification over regions in Euclidean space. Specifically we show the following: 1. Let U be any class of closed regions in Euclidean space that includes all simple polygons. Let C(x, y) be the relation, “Region x is connected to region y,” and let Convex(x) be the property, “Region x is convex.” Then any relation over U that is analytical and invariant under affine transformations is first-order definable in the structure 〈U , C, Convex〉. 2. Let U be as in (1), and let Closer(x, y, z) be the relation “Region x is closer to y than to z.” Then any relation over U that is analytical and invariant under orthogonal transformations is first-order definable in the structure 〈U , Closer〉. 3. Let U be the class of finite unions of intervals in the real line. Then any relation over U that is analytical and invariant under linear transformations is first-order definable in the structure 〈U , Closer〉. 4. If the class of regions is restricted to be polygons with rational vertices, then results analogous to (1-3) hold, substituting “arithmetical relation” for “analytical relation”.