---
tag: KY7SB854VH
kind: concept
name: Face of a convex set
msc: [52A20, 52A05]
sources:
  - url: https://en.wikipedia.org/wiki/Face_(geometry)
---

## Summary

A face of a [[convex set|02AS45T3H9]] $C$ is a convex subset $F$ of $C$ with the following property.
Any segment of $C$ whose inside meets $F$ lies entirely in $F$.
In other words, a face is a convex [[extreme set|WD3RJZC5A7]] of $C$.
For a polytope, the nonempty faces are the vertices, edges and higher-dimensional faces, together with the polytope itself.
The zero-dimensional faces of $C$ are exactly its [[extreme points|3MSGFJM2NX]].

## Explanations

### L1 human

The faces of a cube are the cube itself, its six square sides, its twelve edges, its eight corners, and the empty set.
Each is a flat part of the boundary, or the whole cube, that cannot be entered from outside along a straight line through its inside.
If a segment of the cube passes through the middle of an edge, the segment must run along that edge.
For a disc, the faces are the empty set, the disc and its single boundary points.