Convex sets and Axiom of Choice

TL;DR

This paper explores the logical strength of the statement that every subset of an $ eal$-vector space has a maximal convex subset, revealing its equivalence to various forms of the Axiom of Choice depending on the space's dimension.

Contribution

It establishes the equivalence between the maximal convex subset statement and key axioms like Choice, Countable Choice, and Uniformization in different $ eal$-vector spaces.

Findings

For $ eal^2$, the statement is equivalent to the Axiom of Countable Choice.

For $ eal^3$, it is equivalent to the Axiom of Uniformization.

The strength of the statement varies with the dimension of the space.

Abstract

Under $\mathrm{ZF}$, we show that the statement that every subset of every $\mathbb{R}$-vector space has a maximal convex subset is equivalent to the Axiom of Choice. We also study the strength of the same statement restricted to some specific $\mathbb{R}$-vector spaces. In particular, we show that the statement for $\mathbb{R}^2$ is equivalent to the Axiom of Countable Choice for reals, whereas the statement for $\mathbb{R}^3$ is equivalent to the Axiom of Uniformization. We discuss the statement for some spaces of higher dimensions as well.