# About Balanced-Convex Hull of a Set

Definition. $\newcommand{\bco}{\operatorname{bco}}$Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, deonoted $\bco A$, is the intersection of all balanced-convex subsets of $X$ that contains $A$.

Problem. Let $X$ be a real vector space and let $A\subseteq X$. Then $$\bco A=\left\{\sum_{i=1}^na_ix_i: n\in\mathbb{N},x_i\in A, a_i\in\mathbb{R}\text{ and }\sum_{i=1}^n|a_i|\le1 \right\}.$$

Let us denote the right hand side by $G$. So far, I have shown that $\bco A\subseteq G$. My problem right now is how to show that $G\subseteq \bco A$. Can you please help me... Thanks in advance...

Let $\sum_{i=1}^n a_i \, x_i \in G$ be given. Then, for all $i$, $-x_i, x_i \in \text{bco}A$. Hence $\sum_{i=1}^n |a_i| \, (\text{sign}(a_i) \, x_i) \in \text{bco}A$.