# face of a convex set

Let $C$ be a convex set in $\mathbb{R}^{n}$ (or any topological vector space  ). A face of $C$ is a subset $F$ of $C$ such that

1. 1.

$F$ is convex, and

2. 2.

given any line segment  $L\subseteq C$, if $\operatorname{ri}(L)\cap F\neq\varnothing$, then $L\subseteq F$.

Here, $\operatorname{ri}(L)$ denotes the relative interior of $L$ (open segment of $L$).

This definition formalizes the notion of a face of a convex polygon or a convex polytope and generalizes it to an arbitrary convex set. For example, any point on the boundary of a closed unit disk in $\mathbb{R}^{2}$ is its face (and an extreme point).

Observe that the empty set  and $C$ itself are faces of $C$. These faces are sometimes called improper faces, while other faces are called proper faces.

Remarks. Let $C$ be a convex set.

## References

Title face of a convex set FaceOfAConvexSet 2013-03-22 16:23:08 2013-03-22 16:23:08 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 52A99 ExtremePoint face proper face extreme point improper face