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  "body": "\n## Summary\n\nA subset of a real or complex vector space is balanced, or circled, if it is stable under scaling.\nHere scaling means multiplication by any scalar of absolute value at most $1$.\nIn a topological vector space the balanced neighbourhoods of $0$ form a neighbourhood base at $0$.\nA set that is convex, balanced and absorbing gives a [[seminorm|FP5GVCN8GV]], through the [[Minkowski functional|PAX9SHDHDW]].\n\n## Explanations\n\n### L1 human\n\nA balanced set is symmetric about the origin in a strong sense.\nOver the reals, it contains $-x$ whenever it contains $x$, and it contains the whole segment from $-x$ to $x$.\nOver the complex numbers, it also contains every rotation $e^{i \\theta} x$.\nA disc centred at $0$ is balanced.\nThe interval $[0, 1]$ is not, because it contains $1$ but not $-1$.\n",
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