= Balanced set
{title2=$|\lambda|\leq1\Longrightarrow\lambda B\subseteq B$}
A subset $B$ of a real or complex <vector space> is balanced when
$$
\lambda B\subseteq B\qquad\text{whenever }|\lambda|\leq1.
$$
A nonempty balanced set contains zero, is invariant under multiplication by scalars of modulus one and contains the segment from zero to each of its points. Balance is often called circled in the complex setting.
Every centered norm ball is balanced, by homogeneity of the norm. In $\mathbb R^2$, the union of the two coordinate-axis unit segments is balanced but not convex: it contains $(1,0)$ and $(0,1)$ but not their midpoint. Thus radial scalar closure does not imply closure under arbitrary convex combinations.
The interval $[-1,1]$ is balanced as a subset of $\mathbb R$, but not as a subset of $\mathbb C$, because multiplication by $i$ leaves the real axis. The scalar field therefore matters. Sets that are both convex and balanced are called absolutely convex and are useful for describing locally convex neighborhoods.
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