Convex Combination

A subset A of a vector space V is said to be convex if lambdax+(1-lambda)y for all vectors x,y in A, and all scalars lambda in [0,1]. Via induction, this can be seen to be equivalent to the requirement that lambda_1x_1+...+lambda_nx_n in A for all vectors x_1,x_2,...,x_n in A, and for all scalars lambda_1,lambda_2,...,lambda_n>=0 such that sumlambda_i=1. With the above restrictions on the lambda_i, an expression of the form lambda_1x_1+...+lambda_nx_n is said to be a convex combination of the vectors x_1,x_2,...,x_n.

The set of all convex combinations of vectors in A constitute the convex hull of A so, for example, if x,y in V are two different vectors in the vector space V, then the set of all convex combinations of x and y constitute the line segment between x and y.

This entry contributed by Rasmus Hedegaard

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Hedegaard, Rasmus. "Convex Combination." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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