Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
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.
answered May 11, 2011 at 9:47
-
$\begingroup$ These piecewise linear functions that you describe remind me of "virtual polytopes", of which is known, I believe, that not all of them correspond to convex polytopes. Could you say a bit more about this direction, how to get from the piecewise linear function to the convex polytope? $\endgroup$ Commented Feb 19, 2022 at 21:06