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Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!

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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.

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