1012.5350 (Gen Kimura et al.)

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

Gen Kimura, Koji Nuida

In recent studies, properties of the set of affine maps between two convex sets have been investigated with intensive motivation from quantum physics, but in those preceding works the underlying convex sets were assumed to be compact. In the first part of this article, we establish several mathematical basics to study the set of affine maps between possibly non-compact convex sets, including the definitions of "essential subsets" of the compact closure of a given convex set and a weakened variant of the compact-open topology on the set of affine maps. On the other hand, in the second part, we present a result on separation of simplices and balls (up to affine equivalence) among all compact convex sets in three or lower dimensional Euclidean spaces, based on some characterizing properties of the set of extreme points and the action of affine transformations on it. From a physical viewpoint, this result provides a separation of 2-level quantum mechanical systems from the other possible physical systems. We also give an extension of our result to higher dimensional cases.

View original: http://arxiv.org/abs/1012.5350