\begin{definition}[Definition:Convex Hull]
Let $X$ be a vector space over $\R$. 
Let $U \subseteq X$. 
We define the '''convex hull''' of $U$, written $\map {\operatorname {conv} } U$ by: 
:$\ds \map {\operatorname {conv} } U = \set {\sum_{j \mathop = 1}^n \lambda_j u_j : n \in \N, \, u_j \in U \text { and } \lambda_j \in \R_{> 0} \text { for each } j, \, \sum_{j \mathop = 1}^n \lambda_j = 1}$
\end{definition}