Modified convexity definition on manifolds allowing non-unique minimising geodesics

Question

I am interested in subsets $A \subset M$ of a connected, complete Riemannian manifold $(M,g)$ with the following property: for every $p,q \in A$, at least one minimising geodesic from $p$ to $q$ in $M$ is contained in $A$. Is there a name for this property? It is more general than convexity and weak convexity as defined in Chavel's Riemannian Geometry: A Modern Introduction, since it includes e.g. `bands' on a cylinder.