Balanced set

Remarks

In short, $\lambda A \subseteq A$ whenever $\abs{\lambda} \le 1$. Balanced sets are also called circled. A set that is both convex and balanced is called absolutely convex, or a disk. Example: the closed unit ball $A = \set{x : \norm{x} \le 1}$ of a normed space is balanced, since $\norm{\lambda a} = \abs{\lambda} \norm{a} \le 1$ for $a \in A$ and $\abs{\lambda} \le 1$. Non-example: in $\R$, the interval $[0, 1]$ is not balanced, since $1 \in [0, 1]$ but $(-1) \cdot 1 \notin [0, 1]$. Over $\C$ the condition is stronger: the square $Q = \set{a + b i : \abs{a} \le 1, \abs{b} \le 1}$ is balanced in the real vector space $\C = \R^2$, but not in the complex vector space $\C$, since $1 + i \in Q$ while $e^{i \pi / 4} (1 + i) = \sqrt{2} \, i \notin Q$.