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Convex geometry has recently attracted great attention as a framework toformulate general probabilistic theories. In this framework, convex sets andaffine maps represent the state spaces of physical systems and the possibledynamics, respectively. In the first part of this paper, we present a result onseparation of simplices and balls (up to affine equivalence) among all compactconvex sets in two- and three-dimensional Euclidean spaces, which focuses onthe set of extreme points and the action of affine transformations on it.Regarding the above-mentioned axiomatization of quantum physics, our resultcorresponds to the case of simplest (2-level) quantum system. We also discuss apossible extension to higher dimensions. In the second part, towardsgeneralizations of the framework of general probabilistic theories and severalexisting results including ones in the first part from the case of compact andfinite-dimensional physical systems as in most of the literatures to moregeneral cases, we study some fundamental properties of convex sets and affinemaps that are relevant to the above subject.

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