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A dyadic view of rational convex sets

Gábor Czédli, Miklós Maróti, A. B. Romanowska

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}.

A dyadic view of rational convex sets

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}.

Paper Structure

This paper contains 4 sections, 7 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction and motivation
  2. Barycentric algebras over unital subrings of $\mathbb R$ and the results
  3. Examples and comments
  4. Auxiliary statements and proofs

Key Result

Lemma 2.2

Let $y$ and $x$ be distinct points in $\mathbb R^n$, see Figure figone. Then for each $b$ belonging to the open line segment connecting $y$ and $x$ and for each $p\in I^{o}(R)$, Moreover, $\textup{dist}(y,x) =\textup{dist}(y,b) / p$.

Figures (3)

  • Figure 1: Illustrating Lemma \ref{['barireveAl']} in case $p=1/3$
  • Figure 2: The case $k=1$ and $p=u/v=3/7$
  • Figure 3: Illustrating $\Phi^{\underline{I}^{o}(T)}_{-2/4,\,6/4}(x_0,x_1;y)$

Theorems & Definitions (17)

  • Lemma 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 4.1
  • proof : Proof of Lemma \ref{['iHezTcXc']}
  • ...and 7 more