# Figure out what the faces of convex set are?

I'm trying to understand the concept of faces in convex set. But, I'm stuck with it.

Namely, if I have a convex set such that $C = \{(x,y) \mid x^2 + y^2 \leqslant 1 \}$, then it is clearly convex set. But, I don't know how to find the faces in this set.

Any ideas or explanation would be helpful to me.

• What definition are you using for "face of a convex set"? Is a point on the boundary of the disc a face according to this definition? – Brian Borchers Sep 18 '17 at 2:30
• In good MathJax usage, the entire expression $$C = \{(x,y) \mid x^2 + y^2 \leqslant 1 \}$$ should be within just one pair of dollar signs or double dollar signs. I edited the question accordingly. – Michael Hardy Sep 18 '17 at 2:41

If you are willing to understand what are the faces of a general convex set $C$, you might start to look at what are called the exposed faces of a convex set, which is a notion easy to handle. Exposed faces are a subclass of faces which are particularly easy to define: $F \subset C$ is an exposed face if it is the intersection of the boundary of $C$ together with some hyperplane being tangent to $C$.