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

Gen Kimura, K. Nuida

Research output: Contribution to journalArticle

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 literature to more general cases, we study some fundamental properties of convex sets and affine maps that are relevant to the above subject.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalJournal of Geometry and Physics
Volume86
DOIs
Publication statusPublished - 2014

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ellipsoids
Euclidean geometry
equivalence
balls
physics
geometry

Keywords

  • 2-level quantum system
  • Convex set
  • Ellipsoid
  • Vertex-transitivity

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

Cite this

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