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.]



Support Functions Of 3D Convex Bodies In Spherical Polar Coordinates
For me, the simplest way to figure out whether a function, which is often defined as a function of the unit sphere, is a support function or not is to extend it to all of $\mathbb{R}^n$ as a function homogeneous of degree $1$. Then the function is a support function if and only if it is convex.


You can figure out whether a smooth function of $\theta$ and $\phi$ is a support function or not by finding the formula for this function in terms of the extended function and differentiating twice.


