  The ratio of convexity radius over injectivity radius may be made arbitrarily
small within the class of compact Riemannian manifolds of any fixed dimension
at least two. This is proved using Gulliver's method of constructing manifolds
with focal points but no conjugate points. The approach is suggested by a
characterization of the convexity radius that resembles a classical result of
Klingenberg about the injectivity radius.
