Algebraic closure (convex analysis)

Algebraic closure of a subset ${\displaystyle A}$ of a vector space ${\displaystyle X}$ is the set of all points that are linearly accessible from ${\displaystyle A}$. It is denoted by ${\displaystyle \operatorname {acl} A}$ or ${\displaystyle \operatorname {acl} _{X}A}$.

A point ${\displaystyle x\in X}$ is said to be linearly accessible from a subset ${\displaystyle A\subseteq X}$ if there exists some ${\displaystyle a\in A}$ such that the line segment ${\displaystyle [a,x):=a+[0,1)(x-a)}$ is contained in ${\displaystyle A}$.

Necessarily, ${\displaystyle A\subseteq \operatorname {acl} A\subseteq \operatorname {acl} \operatorname {acl} A\subseteq {\overline {A}}}$ (the last inclusion holds when X is equipped by any vector topology, Hausdorff or not).

The set A is algebraically closed if ${\displaystyle A=\operatorname {acl} A}$. The set ${\displaystyle \operatorname {acl} A\setminus \operatorname {aint} A}$ is the algebraic boundary of A in X.