balanced set

Definition [1, 2, 3, 4]Let $V$ be a vector space over $ℝ$ (or $ℂ$),and let $S$ be a subset of $V$. If $\lambda S\subset S$ for all scalars $\lambda$ suchthat $|\lambda |\le 1$, then $S$ is a balanced set in $V$.The balanced hull of $S$,denoted by $\mathrm{eq}(S)$, is the smallestbalanced set containing $S$.

In the above,$\lambda S=\{\lambda s\mid s\in S\}$,and $|\cdot |$ is the absolute value (in $ℝ$),or the modulus of a complex number (in $ℂ$).

0.0.1 Examples and properties

1. Let $V$ be a normed space with norm $||\cdot ||$. Then the unit ball$\{v\in V\mid ||v||\le 1\}$ is a balanced set.

2. Any vector subspace is a balanced set. Thus, in ${ℝ}^3$, lines and planes passingthrough the origin are balanced sets.

0.0.2 Notes

A balanced set is also sometimes called circled [3].The term balanced evelope is also used for the balanced hull [2].Bourbaki uses the term équilibré [2], c.f. $\mathrm{eq}(A)$above. In [5], a balanced set is defined as above, but with the condition $|\lambda |=1$ instead of $|\lambda |\le 1$.