On affine maps on non-compact convex sets and some characterizations of finite-dimensional solid ellipsoids

TL;DR

This paper investigates affine maps on convex sets, characterizes finite-dimensional solid ellipsoids, and explores their role in general probabilistic theories, focusing on separation properties and extensions to higher dimensions.

Contribution

It provides new characterizations of convex sets like simplices and ellipsoids and extends the framework of probabilistic theories beyond compact, finite-dimensional systems.

Findings

Separation of simplices and balls in 2D and 3D convex sets.

Characterization of affine maps related to quantum systems.

Discussion on extending results to higher dimensions.

Abstract

Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics, respectively. In the first part of this paper, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in two- and three-dimensional Euclidean spaces, which focuses on the set of extreme points and the action of affine transformations on it. Regarding the above-mentioned axiomatization of quantum physics, our result corresponds to the case of simplest (2-level) quantum system. We also discuss a possible extension to higher dimensions. In the second part, towards generalizations of the framework of general probabilistic theories and several existing results including ones in the first part from the case of compact and finite-dimensional physical systems as in most of the literatures to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.