convex combination ConvexCombination 2001-10-19 22:43:33 2006-08-25 08:41:58 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $V$ be some vector space over $\Bbb{R}$. Let $X$ be some set of elements of $V$. Then a {\it convex combination} of elements from $X$ is a linear combination of the form $$\lambda_1 x_1 + \lambda_2 x_2 + \cdots + \lambda_n x_n$$ for some $n > 0$, where each $x_i \in X$, each $\lambda_i \ge 0$ and $\sum_i \lambda_i = 1$. Let ${\rm co}(X)$ be the set of all convex combinations from $X$. We call ${\rm co}(X)$ the {\it convex hull}, or {\it convex envelope}, or {\it convex closure} of $X$. It is a convex set, and is the smallest convex set which contains $X$. A set $X$ is convex if and only if $X = {\rm co}(X)$.