  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.
