Convex sets and Axiom of Choice

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.