nLab absolutely convex subset

Definition

A subset $B$ of a vector space $V$ over $\mathbb{R}$ or $\mathbb{C}$ is absolutely convex if $\lambda x + \mu y \in B$ whenever $x, y \in B$ and ${|\lambda|} + {|\mu|} \le 1$.

Properties

Absolutely convex subsets are closely related to semi-norms. Given a vector space $V$ and an absolutely convex subset $B \subseteq V$, we define $V_B$ to be the linear span of $B$ in $V$. Then let $\mu_B \colon V_B \to \mathbb{R}$ be the Minkowski functional of $B$. That is, $\mu_B$ is given by:

$$\mu_B(v) = \inf\{ t \gt 0 : t v \in B\}$$

Then $\mu_B$ is a semi-norm on $V_B$ and $E_B \coloneqq V_B/\ker \mu_B$ is a normed space.

Related concepts

• convex set
• balanced set