On the geometry of an order unit space

TL;DR

This paper introduces the concepts of skeletons and peripheries to provide a geometric understanding of order unit spaces, exploring their properties and conditions for embedding finite-dimensional spaces.

Contribution

It defines skeletons with heads and peripheries in order unit spaces, offering new geometric insights and conditions for embedding finite-dimensional spaces like b^n.

Findings

Skeletons with heads characterize order unit spaces geometrically.

Peripheries are boundary elements of the positive cone with unit norms.

A condition for embedding b^n as an order unit subspace is established.

Abstract

We introduce the notion of $\mathit{skeleton}$ with a head in a non-zero real vector space. We prove that skeletons with heads describe order unit spaces geometrically. Next, we consider the notion of $\mathit{periphery}$ corresponding to an order unit space which is a part of the skeleton. We note that periphery consists of boundary elements of the positive cone with unit norms. We discuss some elementary properties of the periphery. We also find a condition under which $V$ would contain a copy of $\ell_{\infty}^n$ for some $n \in \mathbb{N}$ as an order unit subspace.