On the axiomatization of convex subsets of Banach spaces

TL;DR

This paper proves that convex-like structures are equivalent to convex subsets of Banach spaces, answering an open question and providing new characterizations of such subsets.

Contribution

It establishes the isometric isomorphism between convex-like structures and convex subsets of Banach spaces, resolving an open problem in the field.

Findings

Convex-like structures are affinely and isometrically isomorphic to convex subsets of Banach spaces.

Brown's algebraic axioms are equivalent to classical axioms of abstract convexity.

Provides a new characterization of convex subsets of Banach spaces.

Abstract

We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces.