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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $V$ be a [[vector space|05EGHEQZAW]] over $\\mathbb{K}$.\nA subset $A \\subseteq V$ is *balanced* if $\\lambda a \\in A$ for every $a \\in A$ and every $\\lambda \\in \\mathbb{K}$ with $\\abs{\\lambda} \\le 1$, where $\\abs{\\lambda}$ is the [[absolute value|00S7C68D51]].\n\n## Remarks\n\nIn short, $\\lambda A \\subseteq A$ whenever $\\abs{\\lambda} \\le 1$.\nBalanced sets are also called *circled*.\nA set that is both convex and balanced is called *absolutely convex*, or a *disk*.\nExample: 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 $\\abs{\\lambda} \\le 1$.\nNon-example: in $\\R$, the interval $[0, 1]$ is not balanced, since $1 \\in [0, 1]$ but $(-1) \\cdot 1 \\notin [0, 1]$.\nOver $\\C$ 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 $e^{i \\pi / 4} (1 + i) = \\sqrt{2} \\, i \\notin Q$.\n",
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