  This paper begins the study of relations between Riemannian geometry and
contact topology in any dimension and continues this study in dimension 3.
Specifically we provide a lower bound for the radius of a geodesic ball in a
contact manifold that can be embedded in the standard contact structure on
Euclidean space, that is on the size of a Darboux ball. The bound is
established with respect to a Riemannian metric compatible with an associated
contact form. In dimension three, it further leads us to an estimate of the
size for a standard neighborhood of a closed Reeb orbit. The main tools are
classical comparison theorems in Riemannian geometry. In the same context, we
also use holomorphic curves techniques to provide a lower bound for the radius
of a PS-tight ball.
