Infinitary primitive positive definability over the real numbers with convex relations

TL;DR

This paper classifies the infinitary primitive positive definability over the real numbers with convex relations into six classes, revealing a geometric and algebraic structure that distinguishes these classes and their logical properties.

Contribution

It provides a complete classification of convex sets based on infinitary primitive positive definability over the reals, introducing a natural partition into six classes with geometric and linear descriptions.

Findings

Six distinct classes of convex sets based on definability.

Elementary geometric and linear descriptions of these classes.

No intermediate clone exists between affine and convex combinations.

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.