Supporting hyperplane

<h2 id="supporting-hyperplane-theorem">Supporting hyperplane theorem</h2>

<p>This <a href="/facts/Theorem/f2Pe1FMD">theorem</a> states that if 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a <a href="/facts/Convex_set/vdAuJRJl">convex set</a> in the <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> 
  
    
      
        X
        =
        
          
            R
          
          
            n
          
        
        ,
      
    
    {\displaystyle X=\mathbb {R} ^{n},}
  
 and 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
 is a point on the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of 
  
    
      
        S
        ,
      
    
    {\displaystyle S,}
  
 then there exists a supporting hyperplane containing 
  
    
      
        
          x
          
            0
          
        
        .
      
    
    {\displaystyle x_{0}.}
  
   If 
  
    
      
        
          x
          
            ∗
          
        
        ∈
        
          X
          
            ∗
          
        
        ∖
        {
        0
        }
      
    
    {\displaystyle x^{*}\in X^{*}\backslash \{0\}}
  
 (
  
    
      
        
          X
          
            ∗
          
        
      
    
    {\displaystyle X^{*}}
  
 is the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
, 
  
    
      
        
          x
          
            ∗
          
        
      
    
    {\displaystyle x^{*}}
  
 is a nonzero linear functional) such that 
  
    
      
        
          x
          
            ∗
          
        
        
          (
          
            x
            
              0
            
          
          )
        
        ≥
        
          x
          
            ∗
          
        
        (
        x
        )
      
    
    {\displaystyle x^{*}\left(x_{0}\right)\geq x^{*}(x)}
  
 for all 
  
    
      
        x
        ∈
        S
      
    
    {\displaystyle x\in S}
  
, then
</p>

H
        =
        {
        x
        ∈
        X
        :
        
          x
          
            ∗
          
        
        (
        x
        )
        =
        
          x
          
            ∗
          
        
        
          (
          
            x
            
              0
            
          
          )
        
        }
      
    
    {\displaystyle H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\}}

<p>defines a supporting hyperplane.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>Conversely, if 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a <a href="/facts/Closed_set/upYnRTUj">closed set</a> with nonempty <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a> such that every point on the boundary has a supporting hyperplane, then 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a convex set, and is the intersection of all its supporting closed half-spaces.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is not convex, the statement of the theorem is not true at all points on the boundary of 
  
    
      
        S
        ,
      
    
    {\displaystyle S,}
  
 as illustrated in the third picture on the right.
</p><p>The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>The forward direction can be proved as a special case of the <a href="/facts/Separating_hyperplane_theorem/9EdtJ3oh">separating hyperplane theorem</a> (see <a href="/facts/Hyperplane_separation_theorem/9EdtJ3oh">the page for the proof</a>). For the converse direction,
</p>
Proof
<p>Define 
  
    
      
        T
      
    
    {\displaystyle T}
  
 to be the intersection of all its supporting closed half-spaces. Clearly 
  
    
      
        S
        ⊂
        T
      
    
    {\displaystyle S\subset T}
  
. Now let 
  
    
      
        y
        ∉
        S
      
    
    {\displaystyle y\not \in S}
  
, show 
  
    
      
        y
        ∉
        T
      
    
    {\displaystyle y\not \in T}
  
.  
</p><p>Let 
  
    
      
        x
        ∈
        
          i
          n
          t
        
        (
        S
        )
      
    
    {\displaystyle x\in \mathrm {int} (S)}
  
, and consider the line segment 
  
    
      
        [
        x
        ,
        y
        ]
      
    
    {\displaystyle [x,y]}
  
. Let 
  
    
      
        t
      
    
    {\displaystyle t}
  
 be the largest number such that 
  
    
      
        [
        x
        ,
        t
        (
        y
        −
        x
        )
        +
        x
        ]
      
    
    {\displaystyle [x,t(y-x)+x]}
  
 is contained in 
  
    
      
        S
      
    
    {\displaystyle S}
  
. Then 
  
    
      
        t
        ∈
        (
        0
        ,
        1
        )
      
    
    {\displaystyle t\in (0,1)}
  
.  
</p><p>Let 
  
    
      
        b
        =
        t
        (
        y
        −
        x
        )
        +
        x
      
    
    {\displaystyle b=t(y-x)+x}
  
, then 
  
    
      
        b
        ∈
        ∂
        S
      
    
    {\displaystyle b\in \partial S}
  
. Draw a supporting hyperplane across 
  
    
      
        b
      
    
    {\displaystyle b}
  
. Let it be represented as a nonzero linear functional 
  
    
      
        f
        :
        
          
            R
          
          
            n
          
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
  
 such that 
  
    
      
        ∀
        a
        ∈
        T
        ,
        f
        (
        a
        )
        ≥
        f
        (
        b
        )
      
    
    {\displaystyle \forall a\in T,f(a)\geq f(b)}
  
. Then since 
  
    
      
        x
        ∈
        
          i
          n
          t
        
        (
        S
        )
      
    
    {\displaystyle x\in \mathrm {int} (S)}
  
, we have 
  
    
      
        f
        (
        x
        )
        >
        f
        (
        b
        )
      
    
    {\displaystyle f(x)>f(b)}
  
. Thus by 
  
    
      
        
          
            
              f
              (
              y
              )
              −
              f
              (
              b
              )
            
            
              1
              −
              t
            
          
        
        =
        
          
            
              f
              (
              b
              )
              −
              f
              (
              x
              )
            
            
              t
              −
              0
            
          
        
        <
        0
      
    
    {\displaystyle {\frac {f(y)-f(b)}{1-t}}={\frac {f(b)-f(x)}{t-0}}<0}
  
, we have 
  
    
      
        f
        (
        y
        )
        <
        f
        (
        b
        )
      
    
    {\displaystyle f(y)<f(b)}
  
, so 
  
    
      
        y
        ∉
        T
      
    
    {\displaystyle y\not \in T}
  
.
</p>

<ul><li><a href="/facts/Support_function/W1k6H5lL">Support function</a></li>
<li><a href="/facts/Supporting_line/FzTIaF47">Supporting line</a> (supporting hyperplanes in 
  
    
      
        
          
            R
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {R} ^{2}}
  
)</li></ul>
<h2 id="notes">Notes</h2>

<h2 id="references--further-reading">References & further reading</h2>
<ul><li>Ostaszewski, Adam (1990). <a href="https://archive.org/details/advancedmathemat0000osta"><i>Advanced mathematical methods</i></a>. Cambridge; New York: Cambridge University Press. p. <a href="https://archive.org/details/advancedmathemat0000osta/page/129">129</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-28964-5.</li></ul>
<ul><li>Giaquinta, Mariano; Hildebrandt, Stefan (1996). <i>Calculus of variations</i>. Berlin; New York: Springer. p. 57. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-50625-X.</li></ul>
<ul><li>Goh, C. J.; Yang, X.Q. (2002). <i>Duality in optimization and variational inequalities</i>. London; New York: Taylor & Francis. p. 13. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-415-27479-6.</li></ul>
<ul><li>Soltan, V. (2021). <i>Support and separation properties of convex sets in finite dimension</i>. Extracta Math. Vol. 36, no. 2, 241-278.</li></ul>

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

<ol>
<li id="fn:1"><p>Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 133. ISBN 978-0-471-18117-0. <a href="978-0-471-18117-0" target="_blank">978-0-471-18117-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 50–51. 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:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 50–51. 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></p></li>
<li id="fn:4"><p>Cassels, John W. S. (1997), An Introduction to the Geometry of Numbers, Springer Classics in Mathematics (reprint of 1959[3] and 1971 Springer-Verlag ed.), Springer-Verlag. <a href="/wiki/John_W._S._Cassels" target="_blank">/wiki/John_W._S._Cassels</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Supporting hyperplane open-in-new

Supporting hyperplane