TITLE: When is a complete fan a normal fan?
QUESTION [7 upvotes]: Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!

REPLY [4 votes]: The characterisation is as follows: There should exist a piecewise linear convex function of the fan, linear at each face of top dimension and having different gradients at all these top-dimensional faces. 
Indeed if you have a convex polytope $P$ with vertices $v_i$ in $\mathbb R^n$ this defines you a collection of linear functions $v^*_i$  on $\mathbb R^{n*}$ and the function $\max_i (v^*_i)$ will be a convex function of the dual fan satisfying the  above properties.