Let $V$ be a real [vector space](/page/Vector%20Space), and let $C \subset V$ be nonempty. Then $C$ is convex if and only if, for every integer $m \geq 1$, every choice of points $c_1,\dots,c_m \in C$, and every choice of scalars $\lambda_1,\dots,\lambda_m \in \mathbb{R}$ satisfying $\lambda_i \geq 0$ for all $i \in \{1,\dots,m\}$ and

\begin{align*} \sum_{i=1}^m \lambda_i = 1, \end{align*}

one has

\begin{align*} \sum_{i=1}^m \lambda_i c_i \in C. \end{align*}