This is inspired by this previous question on physical processes that might give rise to convex hulls.
Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, like paper. We can make the material conform to any surface with nonnegative Gaussian curvature by cutting and folding it. (At least, we can make an arbitrarily good polyhedral approximation.) But if we want to have negative Gaussian curvature, we have to make a cut and glue some extra material in there, which is awkward and cumbersome, so we forbid it. Now we want to perform this gift wrapping as tightly as possible, which suggests using the least amount of material.
Formally, given a set $S$ of points in $\mathbb R^3$, we want to find the surface with minimum area that encloses all the points in $S$, subject to an additional condition that the Gaussian curvature of the surface is nonnegative everywhere. In a comment on the previous question, I conjectured that this would be precisely the convex hull of $S$. But I have no idea if that's actually true, and if so, how to begin proving it.