Balanced Set in Vector Space is Symmetric

Theorem

Let $\GF \in \set {\R, \C}$.

Let $X$ be a vector space over $\GF$.

Let $C \subseteq X$ be a balanced set.

Then $C$ is symmetric.

Proof

Since $C$ is balanced, we have:

$s C \subseteq C$ for all $s \in \C$ with $\cmod s \le 1$.

So in particular, setting $s = -1$:

$-C \subseteq C$

So $C$ is symmetric.