---
tag: 2TDSHJJYS4
kind: concept
name: Balanced set
msc: [46A03, 52A07]
sources:
  - url: https://en.wikipedia.org/wiki/Balanced_set
same_as:
  wikidata: Q778785
  wikipedia: Balanced set
---

## Summary

A subset of a real or complex vector space is balanced, or circled, if it is stable under scaling.
Here scaling means multiplication by any scalar of absolute value at most $1$.
In a topological vector space the balanced neighbourhoods of $0$ form a neighbourhood base at $0$.
A set that is convex, balanced and absorbing gives a [[seminorm|FP5GVCN8GV]], through the [[Minkowski functional|PAX9SHDHDW]].

## Explanations

### L1 human

A balanced set is symmetric about the origin in a strong sense.
Over the reals, it contains $-x$ whenever it contains $x$, and it contains the whole segment from $-x$ to $x$.
Over the complex numbers, it also contains every rotation $e^{i \theta} x$.
A disc centred at $0$ is balanced.
The interval $[0, 1]$ is not, because it contains $1$ but not $-1$.