Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional linearity space whose cones are are in one-to-one correspondence with regular subdivisions of $P$ (see the book by GKZ for a detailed description).
Is there any characterization of those polytopal fans that are the secondary fan of some polytope?
I know that (the pointed part of) secondary fans can be described as the mutual refinement of all simplicial cones spanned by elements in the Gale transform of some polytope. However, I don't know how to get such a description if all I have is the inequalities defining the cones in a polytopal fan (let alone what could go wrong when trying to do that.)
