When is a complete fan a normal fan? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:13:40Z http://mathoverflow.net/feeds/question/64582 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/64582/when-is-a-complete-fan-a-normal-fan When is a complete fan a normal fan? slipstream 2011-05-11T08:55:50Z 2011-05-11T09:47:33Z <p>Is there a characterization for when a complete fan in $\mathbf{R}^n$ is the normal fan of a polytope? Thanks!</p> http://mathoverflow.net/questions/64582/when-is-a-complete-fan-a-normal-fan/64588#64588 Answer by Dmitri for When is a complete fan a normal fan? Dmitri 2011-05-11T09:47:33Z 2011-05-11T09:47:33Z <p>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. </p> <p>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.</p>