Visualization of convex combinations of order 1, 2 and 3 in Geogebra
Convex Combination in 3D with 4 Vectors
Convex Combination in 3D with 4 Vectors



In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. [1]

Definition - Convex Combination


More formally, given a finite number of points   in a real vector space, a convex combination of these points is a point of the form


where the real numbers   satisfy   and  [1]

Remark - outside of convex hull


In the animation of the tetraeder there are examples of points that

  • fulfill  , but
  • violate the condition  .

In case that on   is negative, then linear combination


is a point outside the tetraeder (see animation and consider, when values become negative.

Convex combinations in the plane


If we consider a convex combinations in the plane, then the underlying vector space is the two-dimensional space  . First, we consider convex combinations of two vectors in  . Later we transfer that to infinite dimensional vector spaces of functions and visualize convex combination as GIF animation with Open Source Geogebra. By the condition  , scalars are interdependent. For example, if we define  , then we can set   and  .

Convex combinations as mappings into the vector space

Convex combination of two points as GIF-Animation - generated with geogebra

Convex combination of two functions

Convex combination of two functions visualized in Geogebra

Geogebra: Interaktives Applet - Download: Geogebra-File

Convex Combination as a Mapping


Now we consider the convex combination as a mapping   into the underlying vector space. Due to the fact that the equation can generally represent 1st order convex combinations of 2 vectors   as follows over the mapping  :


Convex Combination of 2 Points


As a particular example, every convex combination of two points lies on the line segment between the points.[1]

Convex Combination of 2 Functions as Vectors


Let   be an interval and as the first function is defined as a polynomial  .


As the second function a trigonometric function   was chosen for the convex combination in the vector space of continuous functions.


The animation above illustrates the convex combination  .

Remark - Vector space of continuous functions


Both functions   and   for all   are elements of the vector space of continuous functions from   to   (i.e.   ).

Remark - Deformation


If the first function   describes the initial shape and   the target shape. A convex combinations of those functions can describe (e.g. in computer graphics) a continuous deformation of an initial shape into a target shape.

Convex combination of more than 2 Points



The figure above, the point   is a convex combination of the three points, while   is not. (  is however an affine combination of the three points, as their affine hull is the entire plane.)]]

Convex combination of n points


A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations.[1]


Linear Combinations and Convex Combinations


There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval   is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither non-negativity nor affinity (i.e., having total integral one).

Convex combinations of convex combinations


In the animation above you can see a convex combination of 2 vectors in the plane or in a function space. If one uses three points then one can create a 1st order convex combination between every two points. We will now consider higher order convex combinations by constructing e.g. a 2nd order convex combination generated from two 1st order convex combinations. Generally from 2 convex combinations of order   you can create a convex combination of order  .

Convex hull


The set of all convex combinations of a given set of vectors is called a convex hull (see also p-convex hull).

Video convex combinations in the plane


Geogebra: Interactive applet - Download:' Geogebra-File

Remarks video about convex combinations of order 1, 2 and 3 in Geogebra


In the video you can see convex combinations of the

  • 1st order between   and   without auxiliary points,
  • 2nd order between   and   with auxiliary point  ,
  • 3rd order between   and   with auxiliary points  ,

Convex combinations as polynomials of t


Convex combinations can be conceived as polynomials where the coefficients come from a vector space   (see also Polynomial Algebra). For example, if one chooses  , one can take a convex combination   to be an element of algebra of polynomial  .

3D convex combination - 1st order


For example, choosing   and  , a 1st order convex combination is defined as follows.


Thus, a 1st order convex combination yields a polynomial of degree 1. with argument  . Represent the convex combination in Geogebra 3D with   (see also Representation of a Straight Line by Direction Vector and Location Vector).

3D convex combination - 2nd order


Choosing again   and   with an auxiliary point  , two 1st order convex combinations yield 2nd order convex combinations.


Represent   as a polynomial   and calculate for   ( ) the coefficients in  .

Bernstein polynomial - order 1


Calculation of the polynomial - order 2


Bernstein polynomial - order 2


Bernstein polynomial - order 3


Convex combination as a mapping


A convex combination can be used to interpolate points   and  . Furthermore, if the auxiliary points  ,....  are given for a convex combination  -th order. The convex combinations can be generally thought of as mapping from the interval   to   as follows:




Convex combinations can also be used to interpolate polynomials. Start first with first order interpolations by interpolating the points with straight lines of the form  . Here, the points   are given data points that are interpolated piecewise using the functions  . Compute from the convex combinations   the functional representation   with  :


Calculation of t as a function of x


Given  . We now compute the corresponding   for the convex combination with the preliminary consideration that   for   and   for  . The following figure takes the linear transformation  .


Calculation of the function value at x


The convex combination


gives the interpolation point of the graph. However, we only need the y-coordinate of the corresponding interpolation point  . So we use the following term:  .

Functional representation


Substituting for   gives the linear interpolation function   over:


Learning Tasks

  • Calculate the coefficients   of the function   with  !
  • Transfer this interpolation to convex combination of order 3 and consider how, depending on the data points, you must choose the two auxiliary points of the interpolation so that the interpolation is differentiable and generates differentiable transitions between the interpolation points in the plot.
  • What geometric properties must auxiliary points between two adjacent interpolation intervals have for differentiability.

Interpolation with convex combination of order 3



Geogebra: Interactive Applet - Download: Geogebra-File


Geogebra: Interactive Applet - Download: Geogebra-File

Develop a mathematical/algebraic description by terms for the following:

  • The green stippled lines are 1st order convex combinations,
  • At the vertices of the open Polygon course, create an angle bisector (constructively, this can be implemented by a rhombus, where two sides and a vertex are defined by two adjacent lines in the polygon course).
  • Create an orthogonal through the connection point of two adjacent lines in the polygon course.
  • Analyze the figure above and determine the next steps for defining the two auxiliary points for a 3rd order convex combination. The procedure is not clear especially at the boundary points of the polygon course. What speaks to your choice of mathematical implementation?

Use altogether methods from linear algebra (e.g. scalar product,... for the vectorial description of the above geometric procedure.

Morphing and the use of convex combinations


In the following section we will consider transformations of images in the context of convex combinations. In Morphing there are different mathematical tools. Here we will only consider aspects in the context of convex combinations.


  • Look at the above GIF animation and first take two different black and white images transform the first image pixel by pixel to the second image by applying a convex combination from gray level values (black=0,...255=white) of a pixel in image 1 to a pixel in image 2 (implementation e.g. in Octave Image Processing v7.3.0 or Octave Image Processing v5.2.0. Note that the convex combinations yields real values of brightness in the gray levels, which must rounded to integer values (e.g. 232.423 to 232 = approximately white). This is necessary due to that the fact that brightness encoding is done with 256 values (Byte).
  • Transfer the procedure to color images only, similarly transferring color values for the primary colors from grayscale values to color values.
  • In the above morphing animation, however, not only static pixel-by-pixel convex combinations are made, but for fixedly defined points, such as eyes, a spatial transformation process also takes place. Consider how, for example, the center of the iris in the eye is spatially shifted from image1 to image2.
  • now connect spatial transformation processes with a color adjustment of the pixels, so that a pixel moves from the location   in the image matrix to   and on the way from   to   the color changes from yellow to blue.



With the following CAS4Wiki commands you can play around with the definition of curves in  

Curves in   and  

Other objects

  • Similarly, a convex combination   of random variables   is a weighted sum (where   satisfy the same constraints as above) of its component probability distributions, often called a finite mixture distribution, with probability density function:
  • Linear Combination
  • A conical combination is a linear combination with nonnegative coefficients. When a point   is to be used as the reference origin for defining displacement vectors, then   is a convex combination of   points   if and only if the zero displacement is a non-trivial conical combination of their   respective displacement vectors relative to  .
  • Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the count of the weights.
  • Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field.

See also



  1. 1.0 1.1 1.2 1.3 Rockafellar, R. Tyrrell (1970), Convex Analysis, Princeton Mathematical Series, vol. 28, Princeton University Press, Princeton, N.J., pp. 11–12, MR 0274683

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