On the Gauss Range of a Closed Convex Set

The aim of this paper is twofold. One one hand we study how the Gauss range of a closed convex subset of the Euclidean space changes when acting on the given set with an affine automorphism. On the other hand we investigate the spherical-convexity of the Gauss range of such a set. Since the Gauss range is not always spherically-convex, we focus our attention on the spherical-convexity of its spherical interior and of its closure as well. Examples of closed convex sets with spherically-convex Gauss ranges are, among others, the epigraphs of suitable lower semicontinuous convex functions.