Centers of disks in Riemannian manifolds

We prove the existence of a center, or continuous selection of a point, in the relative interior of $C^1$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for $k=4=n$. By contrast, for every $n\ge k\ge 6$ there are examples where an equivariant center does not exist. The center can be chosen to agree with any of the classical centers defined on the set of convex compacta in the Euclidean space.