  We examine the relation between oscillatory integral estimates and sublevel
set estimates associated to convex functions. Whilst the former implies the
latter in many cases, the reverse requires additional assumptions. Under finite
(line) type assumptions, Bruna, Nagel & Wainger were able to demonstrate a very
precise control of oscillatory integrals with convex phases via their sublevel
sets. Without the finite type assumption, certain erratic behaviour can force
this precise control to fail (Bak, McMichael, Vance & Wainger). We establish
the same precise control under an alternative qualitative geometric assumption.
