Compositions and Transformations on Spherical Convex Sets

In this paper, we first study some compositions/transformations, such as the scalar-multiplication, spherical addition and canonical transformation, etc., on the unit sphere of Euclidean spaces, and in turn on spherically convex sets in the unit sphere. Then we study valuations on the families of spherically convex sets. In particular, we show that the spherical projections defined recently by Ferreira et al are valuations in some cases.