Dual cone and polar cone

<h2 id="dual-cone">Dual cone</h2>
<h3>In a vector space</h3>
The dual cone C* of a <a href="/facts/Subset/SJGERmbA">subset</a> C in a <a href="/facts/Linear_space/M1pxMLx2">linear space</a> X over the <a href="/facts/Real_numbers/R02gw5Pb">reals</a>, e.g. <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> Rn, with <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> X* is the set

C
          
            ∗
          
        
        =
        
          {
          
            y
            ∈
            
              X
              
                ∗
              
            
            :
            ⟨
            y
            ,
            x
            ⟩
            ≥
            0
            
            ∀
            x
            ∈
            C
          
          }
        
        ,
      
    
    {\displaystyle C^{*}=\left\{y\in X^{*}:\langle y,x\rangle \geq 0\quad \forall x\in C\right\},}

where 
 
 
 
 ⟨
 y
 ,
 x
 ⟩
 
 
 {\displaystyle \langle y,x\rangle }
 
 is the <a href="/facts/Dual_system/2Y8KyofV">duality pairing</a> between X and X*, i.e. 
 
 
 
 ⟨
 y
 ,
 x
 ⟩
 =
 y
 (
 x
 )
 
 
 {\displaystyle \langle y,x\rangle =y(x)}
 
.
C* is always a <a href="/facts/Convex_cone/h1tARfw4">convex cone</a>, even if C is neither <a href="/facts/Convex_set/vdAuJRJl">convex</a> nor a <a href="/facts/Linear_cone/h1tARfw4">cone</a>.

<h3>In a topological vector space</h3>
If X is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> over the real or complex numbers, then the dual cone of a subset C ⊆ X is the following set of continuous linear functionals on X:

C
 
 ′
 
 
 :=
 
 {
 
 f
 ∈
 
 X
 
 ′
 
 
 :
 Re
 ⁡
 
 (
 
 f
 (
 x
 )
 
 )
 
 ≥
 0
 
  for all 
 
 x
 ∈
 C
 
 }
 
 
 
 {\displaystyle C^{\prime }:=\left\{f\in X^{\prime }:\operatorname {Re} \left(f(x)\right)\geq 0{\text{ for all }}x\in C\right\}}
 
,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
which is the <a href="/facts/Polar_set/JY45pfbt">polar</a> of the set -C.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 
No matter what C is, 
 
 
 
 
 C
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }}
 
 will be a convex cone. 
If C ⊆ {0} then 
 
 
 
 
 C
 
 ′
 
 
 =
 
 X
 
 ′
 
 
 
 
 {\displaystyle C^{\prime }=X^{\prime }}
 
.

<h3>In a Hilbert space (internal dual cone)</h3>
Alternatively, many authors define the dual cone in the context of a real <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> (such as Rn equipped with the Euclidean inner product) to be what is sometimes called the internal dual cone.

C
          
            internal
          
          
            ∗
          
        
        :=
        
          {
          
            y
            ∈
            X
            :
            ⟨
            y
            ,
            x
            ⟩
            ≥
            0
            
            ∀
            x
            ∈
            C
          
          }
        
        .
      
    
    {\displaystyle C_{\text{internal}}^{*}:=\left\{y\in X:\langle y,x\rangle \geq 0\quad \forall x\in C\right\}.}

<h3>Properties</h3>
Using this latter definition for C*, we have that when C is a cone, the following properties hold:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>A non-zero vector y is in C* if and only if both of the following conditions hold:</li></ul>
<ol><li>y is a <a href="/facts/Surface_normal/jhtLIhcu">normal</a> at the origin of a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> that <a href="/facts/Supporting_hyperplane/RyA2XtaK">supports</a> C.</li>
<li>y and C lie on the same side of that supporting hyperplane.</li></ol>
<ul><li>C* is <a href="/facts/Closed_set/upYnRTUj">closed</a> and convex.</li>
<li>
 
 
 
 
 C
 
 1
 
 
 ⊆
 
 C
 
 2
 
 
 
 
 {\displaystyle C_{1}\subseteq C_{2}}
 
 implies 
 
 
 
 
 C
 
 2
 
 
 ∗
 
 
 ⊆
 
 C
 
 1
 
 
 ∗
 
 
 
 
 {\displaystyle C_{2}^{*}\subseteq C_{1}^{*}}
 
.</li>
<li>If C has nonempty interior, then C* is pointed, i.e. C* contains no line in its entirety.</li>
<li>If C is a cone and the closure of C is pointed, then C* has nonempty interior.</li>
<li>C** is the closure of the smallest convex cone containing C (a consequence of the <a href="/facts/Hyperplane_separation_theorem/9EdtJ3oh">hyperplane separation theorem</a>)</li></ul>
<h2 id="self-dual-cones">Self-dual cones</h2>
A cone C in a vector space X is said to be self-dual if X can be equipped with an <a href="/facts/Inner_product/HoyjElEy">inner product</a> ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to C.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 
Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual. 
This is slightly different from the above definition, which permits a change of inner product. 
For instance, the above definition makes a cone in Rn with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in Rn is equal to its internal dual.
The nonnegative <a href="/facts/Orthant/3bkJde4G">orthant</a> of Rn and the space of all <a href="/facts/Positive_semidefinite_matrix/Hbr6AuPS">positive semidefinite matrices</a> are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). 
So are all cones in R3 whose base is the convex hull of a regular polygon with an odd number of vertices. 
A less regular example is the cone in R3 whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.

<h2 id="polar-cone">Polar cone</h2>

For a set C in X, the polar cone of C is the set<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

C
          
            o
          
        
        =
        
          {
          
            y
            ∈
            
              X
              
                ∗
              
            
            :
            ⟨
            y
            ,
            x
            ⟩
            ≤
            0
            
            ∀
            x
            ∈
            C
          
          }
        
        .
      
    
    {\displaystyle C^{o}=\left\{y\in X^{*}:\langle y,x\rangle \leq 0\quad \forall x\in C\right\}.}

It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C*.
For a closed convex cone C in X, the polar cone is equivalent to the <a href="/facts/Polar_set/JY45pfbt">polar set</a> for C.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bipolar_theorem/MtShefAp">Bipolar theorem</a></li>
<li><a href="/facts/Polar_set/JY45pfbt">Polar set</a></li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Vladimir_Boltyansky/M05GyYD9">Boltyanski, V. G.</a>; Martini, H.; Soltan, P. (1997). Excursions into combinatorial geometry. New York: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-61341-2.{{cite book}}: CS1 maint: publisher location (link)</li>
<li>Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-415-27479-6.</li>
<li>Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1584888666. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/144216834">144216834</a>.</li>
<li>Ramm, A.G. (2000). Shivakumar, P.N.; Strauss, A.V. (eds.). Operator theory and its applications. Providence, R.I.: American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1990-9.</li>
<li><a href="/facts/Helmut_H._Schaefer/xorHTDBE">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-7155-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/840278135">840278135</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schaefer & Wolff 1999, pp. 215–222. - Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. <a href="https://search.worldcat.org/oclc/840278135" target="_blank">https://search.worldcat.org/oclc/840278135</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 978-0-521-83378-3. Retrieved October 15, 2011. <a href="978-0-521-83378-3" target="_blank">978-0-521-83378-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6. <a href="978-0-691-01586-6" target="_blank">978-0-691-01586-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0. <a href="978-3-540-32696-0" target="_blank">978-3-540-32696-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Dual cone and polar cone open-in-new

Dual cone and polar cone