What are the known sufficient conditions,analagous to the planar curvature condition, in terms of functions of theta and phi, on the support function h(theta,phi) of a surface in 3D which imply it is the boundary of a convex body?

[The origin of this question is an elementary approach to exploring 3D (convex) sets of constant width (and their volumes using the Divergence Theorem) in advance of the anniversary of the Blaschke-Lebesgue (1914) theorem.]