Three theorems for Polyhedra, Polytopes, and Cones

Question

Are there readable proofs of the following theorems?

1. A polytope (bounded polyhedron) is the convex hull of a finite set of points.

2. A polyhedral cone is generated by a finite set of vectors. That is for any $m \times n$ matrix $A$, there exists a finite set $X$ such that $\{x=\sum\lambda_ix_i, x_i\in X, \lambda_i\ge 0\}=\{x: Ax\le 0\}$.

3. A polyhedron $P:\{x: Ax\le b\}$ can be written as the Minkowski sum of a polytope $Q$ and a cone $C$ $\left(\text{i.e.}, P=Q+C=\{x+y, x\in Q, y\in C\}\right)$.

I was wondering if someone could show me any proof of one of three above theorems.

Answer 1

Perhaps you are thinking of

Ziegler, Günter M. Lectures on Polytopes. Vol. 152. Springer Science & Business Media, 1995. (Springer link.)

whose first chapter is on basic properties: Polytopes, Polyhedra, and Cones. Theorem 1.1 on p.29 is your #1. Theorem 1.3 on p.30 is your #2.

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