Convex sets and Axiom of Choice

Yasuo Yoshinobu

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arXiv · 2602.01739

Convex sets and Axiom of Choice

Yasuo Yoshinobu

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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.

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BibTeX RIS

Yasuo Yoshinobu. 2026-02-02. Convex sets and Axiom of Choice. https://arxiv.org/abs/2602.01739

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