  We consider spaces of multivariate splines defined on a particular type of
simplicial partitions that we call (generalized) oranges. Such partitions are
composed of a finite number of maximal faces with exactly one shared medial
face. We reduce the problem of finding the dimension of splines on oranges to
computing dimensions of splines on simpler, lower-dimensional partitions that
we call projected oranges. We use both algebraic and Bernstein-B\'ezier tools.
