# How to prove that faces of a polytope are polytopes

How do you prove that each face of a polytope is also a polytope? I think it may be through induction.

• What is definition of a polytope ? Aug 17 '17 at 7:55
• The definition of polytope is the convex hull of finite set of points. @HKLee Aug 17 '17 at 11:24
• I have one more question. What is the definition of face ? Aug 17 '17 at 12:22
• Let K be a subset of an n-dimensional Euclidean space, then the face, F, is the set x in K such that inner product <x, u> is equal to an alpha. @HKLee Aug 17 '17 at 12:55

Here by polytope I mean (as you) the convex hull of finitely many points. By a (convex) polyhedron I mean a subset of $$\mathbb R^d$$ defined by finitely many affine-linear inequalities. By a face of a polytope or polyhedron $$P$$ I mean the part of $$P$$ where some linear functional is maximized. (This is almost what you say in "the face, F, is the set x in K such that inner product $$$$ is equal to an alpha", except one needs to assume that alpha is the maximum value taken by $$$$ on $$P$$)