Convex Hull of a balanced set is balanced

Question

A set $A$ is balanced if and only if $\alpha A \subseteq A$ for any scalar $\alpha \in [-1,1].$

conv$A = \{ \lambda a + (1-\lambda)b: a,b\in A, \lambda \in [0,1] \}.$

Show that if $A$ is a balanced subset of a vector space $V,$ then $\text{conv}(A)$ is balanced.

My attempt: Let $x \in \text{conv}(A)$ and $\lambda \in [-1,1].$ By the definition of $\text{conv}(A),$ there exists $\mu \in [0,1],$ $a,b\in A$ such that $x = \mu a + (1-\mu)b.$ It follows that $\lambda x = \lambda \mu a + \lambda (1-\mu) b.$ Since $\lambda \mu, \lambda (1-\mu) \in [-1,1],$ we have $\lambda x \in \text{conv}(A).$

Question: Where did I use the condition that $A$ is balanced?

Answer 1

Let $x\in $Conv$A.$ Then there are $a,b\in A$ such that $x=ta+(1-t)b$ for $0\le t\le 1.$

If $-1\le \lambda\le 1$ we have then $\lambda x=t\lambda a+(1-t)\lambda b.$

But $A$ is balanced so $\lambda a,\lambda b\in A,\ $ which implies that $\lambda x\in $Conv$A$ and so Conv$A$ is balanced.

Answer 2

$\lambda(-a)+(1-\lambda)(-b)$ is in conv$A$ .