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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Normal fan</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically <a href="Convex_geometry" title="Convex geometry">convex geometry</a>, the <b>normal fan</b> of a <a href="Convex_polytope" title="Convex polytope">convex polytope</a> <i>P</i> is a <a href="Polyhedral_complex#Fans" title="Polyhedral complex">polyhedral fan</a> that is <a href="Dual_polytope" class="mw-redirect" title="Dual polytope">dual</a> to <i>P</i>. Normal fans have applications to <a href="Polyhedral_combinatorics" title="Polyhedral combinatorics">polyhedral combinatorics</a>, <a href="Linear_programming" title="Linear programming">linear programming</a>, <a href="Tropical_geometry" title="Tropical geometry">tropical geometry</a>, <a href="Toric_geometry" class="mw-redirect" title="Toric geometry">toric geometry</a> and other areas of mathematics.
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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Given a convex polytope <i>P</i> in <b>R</b><sup><i>n</i></sup>, the normal fan <i>N</i><sub><i>P</i></sub> of <i>P</i> is a polyhedral fan in the <a href="Dual_space" title="Dual space">dual space</a>, (<b>R</b><sup><i>n</i></sup>)* whose <a href="Cones" class="mw-redirect" title="Cones">cones</a> consist of the <b>normal cone</b> <i>C</i><sub><i>F</i></sub> to each face <i>F</i> of <i>P</i>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{P}=\{C_{F}\}_{F\in \operatorname {face} (P)}.}">
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<annotation encoding="application/x-tex">{\displaystyle N_{P}=\{C_{F}\}_{F\in \operatorname {face} (P)}.}</annotation>
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</math></span><img src="./a574d6ea6cb5c8547e38644d1dee116b48099877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.386ex; height:3.176ex;" alt="{\displaystyle N_{P}=\{C_{F}\}_{F\in \operatorname {face} (P)}.}" loading="lazy"></span></dd></dl>
<p>Each normal cone <i>C</i><sub><i>F</i></sub> is defined as the set of linear functionals <i>w</i> such that the set of points <i>x</i> in <i>P</i> that maximize <i>w</i>(<i>x</i>) contains <i>F</i>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{F}=\{w\in (\mathbb {R} ^{n})^{*}\mid F\subseteq \operatorname {argmax} _{x\in P}w(x)\}.}">
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<annotation encoding="application/x-tex">{\displaystyle C_{F}=\{w\in (\mathbb {R} ^{n})^{*}\mid F\subseteq \operatorname {argmax} _{x\in P}w(x)\}.}</annotation>
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</math></span><img src="./27ff90a5763c4b593baa373f266251cfbda7b606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.492ex; height:2.843ex;" alt="{\displaystyle C_{F}=\{w\in (\mathbb {R} ^{n})^{*}\mid F\subseteq \operatorname {argmax} _{x\in P}w(x)\}.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<ul><li><i>N</i><sub><i>P</i></sub> is a <i>complete fan</i>, meaning the union of its cones is the whole space, (<b>R</b><sup><i>n</i></sup>)*.</li>
<li>If <i>F</i> is a face of <i>P</i> of dimension <i>d</i>, then its normal cone <i>C</i><sub><i>F</i></sub> has dimension <i>n</i> – <i>d</i>. The normal cones to vertices of <i>P</i> are full dimensional. If <i>P</i> has full dimension, the normal cones to the facets of <i>P</i> are the rays of <i>N</i><sub><i>P</i></sub> and the normal cone to <i>P</i> itself is <i>C</i><sub><i>P</i></sub> = {0}, the zero cone.</li>
<li>The <a href="Affine_span" class="mw-redirect" title="Affine span">affine span</a> of face <i>F</i> of <i>P</i> is <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to the linear span of its normal cone, <i>C</i><sub><i>F</i></sub>.</li>
<li>The correspondence between faces of <i>P</i> and cones of <i>N</i><sub><i>P</i></sub> reverses inclusion, meaning that for faces <i>F</i> and <i>G</i> of <i>P</i>,</li></ul>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subseteq G\quad \Leftrightarrow \quad C_{F}\supseteq C_{G}.}">
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<annotation encoding="application/x-tex">{\displaystyle F\subseteq G\quad \Leftrightarrow \quad C_{F}\supseteq C_{G}.}</annotation>
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</math></span><img src="./7528686eb48394e09fb4b405c626987bc924eeb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.981ex; height:2.509ex;" alt="{\displaystyle F\subseteq G\quad \Leftrightarrow \quad C_{F}\supseteq C_{G}.}" loading="lazy"></span></dd></dl></dd></dl>
<ul><li>Since <i>N</i><sub><i>P</i></sub> is a fan, the <a href="Intersection" title="Intersection">intersection</a> of any two of its cones is also a cone in <i>N</i><sub><i>P</i></sub>. For faces <i>F</i> and <i>G</i> of <i>P</i>,</li></ul>
<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{F}\cap C_{G}=C_{H}}">
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<annotation encoding="application/x-tex">{\displaystyle C_{F}\cap C_{G}=C_{H}}</annotation>
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</math></span><img src="./c2474034584607146ac1b3882dfdbfccd94562aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.345ex; height:2.509ex;" alt="{\displaystyle C_{F}\cap C_{G}=C_{H}}" loading="lazy"></span></dd></dl></dd>
<dd>where <i>H</i> is the smallest face of <i>P</i> that contains both <i>F</i> and <i>G</i>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<ul><li>If polytope <i>P</i> is thought of as the <a href="Feasible_region" title="Feasible region">feasible region</a> of a <a href="Linear_programming" title="Linear programming">linear program</a>, the normal fan of <i>P</i> 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 <i>w</i> has solution set <i>F</i> if and only if <i>w</i> is in the <a href="Relative_interior" title="Relative interior">relative interior</a> of the cone <i>C</i><sub><i>F</i></sub>.</li>
<li>If polytope <i>P</i> has the <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a> in its <a href="Interior_(topology)" title="Interior (topology)">interior</a>, then the normal fan of <i>P</i> can be constructed from the <a href="Dual_polyhedron#Polar_reciprocation" title="Dual polyhedron">polar dual</a> of <i>P</i> by taking the cone over each face of the dual polytope, <i>P</i>°.</li>
<li>For <i>f</i> a polynomial in <i>n</i> variables with coefficients in <b>C</b>, the <a href="Tropical_geometry#Tropical_polynomials" title="Tropical geometry">tropical hypersurface</a> of <i>f</i> is supported on a subfan of the normal fan of the <a href="Newton_polytope" title="Newton polytope">Newton polytope</a> <i>P</i> of <i>f</i>. In particular, the tropical hypersurface is supported on the cones in <i>N</i><sub><i>P</i></sub> of dimension less than <i>n</i>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFZiegler1995" class="citation cs2"><a href="G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a> (1995), <i>Lectures on Polytopes</i>, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94365-X</bdi></cite>.</li></ul>
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