Convex sets and Axiom of Choice
Yasuo Yoshinobu
TL;DR
The paper investigates how maximal convex subsets in ${\mathbb R}$-vector spaces relate to the Axiom of Choice, introducing $\mathrm{MCV}(V)$ and its global form $\mathrm{MCV}$. It establishes that $\mathrm{MCV}$ is equivalent to AC under ZF, and it identifies sharp equivalences in low dimensions: $\mathrm{MCV}(2)$ is equivalent to $\mathrm{CC}_{\mathbb R}$ and $\mathrm{MCV}(3)$ is equivalent to $\mathrm{Unif}_{\mathbb R}$. A robust framework of convex filtrations and faces is developed to construct maximal convex subsets, enabling reductions of CC$_{\mathbb R}$ and Unif$_{\mathbb R}$ to $\mathrm{MCV}(2)$ and $\mathrm{MCV}(3)$, respectively. The work further extends to higher dimensions, connecting $\mathrm{MCV}$ to fragments like $\mathrm{SC}$ and exploring dimension–cardinality interactions, while also addressing combinatorial applications via Moore-type theorems for graphs on surfaces. Overall, the paper delineates a precise hierarchy between maximal-convex-subset principles and classical choice principles, with structured pathways for future analysis in infinite dimensions.
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.