A dyadic view of rational convex sets

TL;DR

This paper explores the algebraic structure of convex sets in real vector spaces under the arithmetic mean operation, establishing conditions for their isomorphism based on automorphisms of the underlying affine space.

Contribution

It characterizes when convex subsets induce isomorphic algebraic structures via affine automorphisms, extending to barycentric algebras over subrings.

Findings

Isomorphism of groupoids corresponds to affine automorphisms of the space.

Results apply to convex sets of the same dimension with at least one bounded or over rationals.

Generalization to barycentric algebras over subrings of F.

Abstract

Let F be a subfield of the field R of real numbers. Equipped with the binary arithmetic mean operation, each convex subset C of F^n becomes a commutative binary mode, also called idempotent commutative medial (or entropic) groupoid. Let C and C' be convex subsets of F^n. Assume that they are of the same dimension and at least one of them is bounded, or F is the field of all rational numbers. We prove that the corresponding idempotent commutative medial groupoids are isomorphic iff the affine space F^n over F has an automorphism that maps C onto C'. We also prove a more general statement for the case when C,C'\subseteq F^n are considered barycentric algebras over a unital subring of F that is distinct from the ring of integers. A related result, for a subring of R instead of a subfield F, is given in \cite{rczgaroman2}.