  An algebraic variety is said to have the $A_k$-property if any $k$ points are
contained in some common affine open neighbourhood. A theorem of W{\l}odarczyk
states that a normal variety has the $A_2$-property if and only if it admits a
closed embedding into a toric variety. Spherical varieties can be regarded as a
generalization of toric varieties, but they do not have the $A_2$-property in
general. We provide a combinatorial criterion for the $A_k$-property of
spherical varieties by combining the theory of bunched rings with the Luna-Vust
theory of spherical embeddings.
