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

Uniform Oscillatory Integral estimates for Convex Phases via Sublevel Set estimates