# Normal fan

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{{short description|Structure in convex geometry}}
{{Multiple issues|{{more footnotes|date=June 2019}}{{refimprove|date=June 2019}}}}

In [mathematics](/source/mathematics), specifically [convex geometry](/source/convex_geometry), the '''normal fan''' of a [convex polytope](/source/convex_polytope) ''P'' is a [polyhedral fan](/source/Polyhedral_complex) that is [dual](/source/Dual_polytope) to ''P''.  Normal fans have applications to [polyhedral combinatorics](/source/polyhedral_combinatorics), [linear programming](/source/linear_programming), [tropical geometry](/source/tropical_geometry), [toric geometry](/source/toric_geometry) and other areas of mathematics.

==Definition==
Given a convex polytope ''P'' in '''R'''<sup>''n''</sup>, the normal fan ''N''<sub>''P''</sub> of ''P'' is a polyhedral fan in the [dual space](/source/dual_space), ('''R'''<sup>''n''</sup>)* whose [cones](/source/cones) consist of the '''normal cone''' ''C''<sub>''F''</sub> to each face ''F'' of ''P'',
:<math>N_P = \{C_F\}_{F \in \operatorname{face}(P)}.</math>
Each normal cone ''C''<sub>''F''</sub> is defined as the set of linear functionals ''w'' such that the set of points ''x'' in ''P'' that maximize ''w''(''x'') contains ''F'',
:<math>C_F = \{w \in (\mathbb{R}^n)^* \mid F \subseteq \operatorname{argmax}_{x \in P} w(x) \}.</math>

==Properties==
* ''N''<sub>''P''</sub> is a ''complete fan'', meaning the union of its cones is the whole space, ('''R'''<sup>''n''</sup>)*.
* If ''F'' is a face of ''P'' of dimension ''d'', then its normal cone ''C''<sub>''F''</sub> has dimension ''n'' – ''d''.  The normal cones to vertices of ''P'' are full dimensional.  If ''P'' has full dimension, the normal cones to the facets of ''P'' are the rays of ''N''<sub>''P''</sub> and the normal cone to ''P'' itself is ''C''<sub>''P''</sub> = {0}, the zero cone.
* The [affine span](/source/affine_span) of face ''F'' of ''P'' is [orthogonal](/source/orthogonal) to the linear span of its normal cone, ''C''<sub>''F''</sub>.
* The correspondence between faces of ''P'' and cones of ''N''<sub>''P''</sub> reverses inclusion, meaning that for faces ''F'' and ''G'' of ''P'',
::<math>F \subseteq G \quad \Leftrightarrow \quad C_F \supseteq C_G.</math>
* Since ''N''<sub>''P''</sub> is a fan, the [intersection](/source/intersection) of any two of its cones is also a cone in ''N''<sub>''P''</sub>.  For faces ''F'' and ''G'' of ''P'',
::<math>C_F \cap C_G = C_H</math>
:where ''H'' is the smallest face of ''P'' that contains both ''F'' and ''G''.

==Applications==
* If polytope ''P'' is thought of as the [feasible region](/source/feasible_region) of a [linear program](/source/linear_programming), the normal fan of ''P'' partitions the space of objective functions based on the solution set to the linear program defined by each.  The linear program in which the goal is to maximize linear objective function ''w'' has solution set ''F'' if and only if ''w'' is in the [relative interior](/source/relative_interior) of the cone ''C''<sub>''F''</sub>.
* If polytope ''P'' has the [origin](/source/Origin_(mathematics)) in its [interior](/source/Interior_(topology)), then the normal fan of ''P'' can be constructed from the [polar dual](/source/Dual_polyhedron) of ''P'' by taking the cone over each face of the dual polytope, ''P''°.
* For ''f'' a polynomial in ''n'' variables with coefficients in '''C''', the [tropical hypersurface](/source/Tropical_geometry) of ''f'' is supported on a subfan of the normal fan of the [Newton polytope](/source/Newton_polytope) ''P'' of ''f''.  In particular, the tropical hypersurface is supported on the cones in ''N''<sub>''P''</sub> of dimension less than ''n''.

==References==
{{refbegin}}
*{{citation
 | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler
 | isbn = 0-387-94365-X
 | publisher = Springer-Verlag
 | series = Graduate Texts in Mathematics
 | title = Lectures on Polytopes
 | volume = 152
 | year = 1995}}.
{{refend}}
Category:Geometric objects

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Adapted from the Wikipedia article [Normal fan](https://en.wikipedia.org/wiki/Normal_fan) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Normal_fan?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.