In convex analysis, a convex cone can be viewed as being like a one-sided version of a subspace. (And the polar cone is analogous to the orthogonal complement. It's a nice analogy.)
A smooth manifold has a tangent space at every point. Is there an analogous type of mathematical structure which is sort of like a smooth manifold, but which has a convex cone at every point? (Or can be approximated in some sense by a convex cone at every point?)
If so then what is the name of this type of object?