Algebraic closure (convex analysis)

Algebraic closure of a subset of a vector space is the set of all points that are linearly accessible from. It is denoted by or.
A point is said to be linearly accessible from a subset if there exists some such that the line segment is contained in.
Necessarily, .
The set A is algebraically closed if.
The set is the algebraic boundary of A in X.

Examples

The set of rational numbers is algebraically closed but is not algebraically open
If then
. In particular, the algebraic closure need not be algebraically closed.
Here,.
However, for every finite-dimensional convex set A.
Moreover, a convex set is algebraically closed if and only if its complement is algebraically open.