Convex combination

<h2 id="formal-definition">Formal definition</h2>
<p>More formally, given a finite number of points 
  
    
      
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
      
    
    {\displaystyle x_{1},x_{2},\dots ,x_{n}}
  
 in a <a href="/facts/Real_vector_space/M1pxMLx2">real vector space</a>, a convex combination of these points is a point of the form
</p>

α
          
            1
          
        
        
          x
          
            1
          
        
        +
        
          α
          
            2
          
        
        
          x
          
            2
          
        
        +
        ⋯
        +
        
          α
          
            n
          
        
        
          x
          
            n
          
        
      
    
    {\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}}

<p>where the real numbers 
  
    
      
        
          α
          
            i
          
        
      
    
    {\displaystyle \alpha _{i}}
  
 satisfy 
  
    
      
        
          α
          
            i
          
        
        ≥
        0
      
    
    {\displaystyle \alpha _{i}\geq 0}
  
 and 
  
    
      
        
          α
          
            1
          
        
        +
        
          α
          
            2
          
        
        +
        ⋯
        +
        
          α
          
            n
          
        
        =
        1.
      
    
    {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.}
  
<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>As a particular example, every convex combination of two points lies on the <a href="/facts/Line_segment/V8E0M4qk">line segment</a> between the points.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>A set is <a href="/facts/Convex_set/vdAuJRJl">convex</a> if it contains all convex combinations of its points.
The <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of a given set of points is identical to the set of all their convex combinations.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval 
  
    
      
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle [0,1]}
  
 is convex but generates the real-number line under linear combinations. Another example is the convex set of <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a>, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).
</p>
<h2 id="other-objects">Other objects</h2>
<ul><li>A random variable 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is said to have an 
  
    
      
        n
      
    
    {\displaystyle n}
  
-component <a href="/facts/Mixture_distribution/BMdA5kTI">finite mixture distribution</a> if its <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is a convex combination of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 so-called component densities.</li></ul>
<h2 id="related-constructions">Related constructions</h2>
<p class="note">Further information: <a href="/facts/Linear_combination/CVjL6kuN">Linear combination § Affine, conical, and convex combinations</a></p>
<ul><li>A <a href="/facts/Conical_combination/41KGtqdd">conical combination</a> is a linear combination with nonnegative coefficients. When a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is to be used as the reference origin for defining <a href="/facts/Displacement_(vector)/jkmEfS3C">displacement vectors</a>, then 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a convex combination of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 points 
  
    
      
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
      
    
    {\displaystyle x_{1},x_{2},\dots ,x_{n}}
  
 if and only if the zero displacement is a non-trivial <a href="/facts/Conical_combination/41KGtqdd">conical combination</a> of their 
  
    
      
        n
      
    
    {\displaystyle n}
  
 respective displacement vectors relative to 
  
    
      
        x
      
    
    {\displaystyle x}
  
.</li>
<li><a href="/facts/Weighted_mean/VHyJfPxb">Weighted means</a> are functionally the same as convex combinations, but they use a different notation. The coefficients (<a href="/facts/Weight_function/YNJL7o99">weights</a>) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the sum of the weights.</li>
<li><a href="/facts/Affine_combination/LCGkRJvC">Affine combinations</a> are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>.</li></ul>
<h2 id="see-also">See also</h2>

Wikiversity has learning resources about <i>Convex combination</i>

<ul><li><a href="/facts/Affine_hull/4fhyHKoa">Affine hull</a></li>
<li><a href="/facts/Carath%C3%A9odory%27s_theorem_(convex_hull)/QGyEIEPv">Carathéodory's theorem (convex hull)</a></li>
<li><a href="/facts/Simplex/0FCfNRN8">Simplex</a></li>
<li><a href="/facts/Barycentric_coordinate_system_(mathematics)/ymlRaudU">Barycentric coordinate system</a></li>
<li><a href="/facts/Convex_space/vdAuJRJl">Convex space</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.geogebra.org/m/prkxbqxx">Convex sum/combination with a triangle</a> - interactive illustration</li>
<li><a href="https://www.geogebra.org/m/ekyrkytj">Convex sum/combination with a hexagon</a> - interactive illustration</li>
<li><a href="https://www.geogebra.org/m/asfdevce">Convex sum/combination with a tetraeder</a> - interactive illustration</li></ul>

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

<ol>
<li id="fn:1"><p>Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683 <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Convex combination open-in-new

Convex combination