Infinitary primitive positive definability over the real numbers with convex relations

Abstract

On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitary primitively positively definable relations are a natural intermediate concept which extends primitive positive definability by infinite conjunctions. We consider for every convex set $S\subset \mathbb{R}^n$ the structure of the real numbers $\mathbb{R}$ with addition, scalar multiplication, constants, and additionally the relation $S$. We prove that depending on $S$, the set of all relations with an infinitary primitive positive definition in this structure equals one out of six possible sets. This dependency gives a natural partition of the convex sets into six nonempty classes. We also give an elementary geometric description of the classes and a description in terms of linear maps. The classification also implies that there is no locally closed clone between the clone of affine combinations and the clone of convex combinations.