Balanced set

A set $U$ in a real or complex vector space $X$ such that $x \in U$ and $|\lambda| \le 1$ imply $\lambda x \in U$. An example of a balanced set is given by the unit ball in a normed vector space and, generally, by a neighbourhood $U$ of zero in a base of neighbourhoods of zero in a topological vector space. These neighbourhoods of zero are moreover absorbing, i.e. such that for any $x \in X$ there exists an $\alpha > 0$ such that $x \in \lambda U$ for $|\lambda| \ge \alpha$. If $U$ is a convex, absorbing and balanced set, then the functional $p_U(x) = \inf\{|\lambda| : x \in \lambda U\}$ is a semi-norm, i.e. it has the properties $$ p_U(x+y) \le p_U(x) + p_U(y)\ ,\ \ p_U(\lambda x) = |\lambda| p(x) \ . $$

A balanced set is also called centred.

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The functional $p_U$ mentioned above is also called the Minkowski functional of the convex, absorbing and balanced set $U$.

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