Characterizing the radial sum for star bodies

In this paper we prove two theorems characterizing the radial sum of star bodies. By doing so we demonstrate an interesting phenomenon: essentially the same conditions, on two different spaces, can uniquely characterize very different operations. In our first theorem we characterize the radial sum by its induced homothety, and our list of assumptions is identical to the assumptions of the corresponding theorem which characterizes the Minkowski sum for convex bodies. In our second theorem give a different characterization from a short list of natural properties, without assuming the homothety has any specific form. For this theorem one has to add an assumption to the corresponding theorem for convex bodies, as we demonstrate by a simple example.