Connection/Consistency Between Different Definitions of Polyhedron? And How Does This Proof Apply to This Definition? My textbook gives the following definition of a polyhedron:

A polyhedron is defined as the solution set of a finite number of linear equalities and inequalities:
$$\mathcal{P} = \{ x \mid a^T_j x \le b_j, \ j = 1, \dots , m, \ c_j^T x = d_j, \ j = 1, \dots , p \}$$
A polyhedron is thus the intersection of a finite number of halfspaces and hyperplanes.  Affine set (e.g., subspaces, hyperplanes, lines), rays, line segments, and halfspaces are all polyhedra. It is easily shown that polyhedra are convex set.

I was trying to find a proof of the fact that polyhedra are convex sets, and so I came across this:

Let $\mathrm{\mathbf{a}}$ be a vector and let $b$ a scalar. Suppose that $\mathrm{\mathbf{x}}$ and $\mathrm{\mathbf{y}}$ satisfy $\mathrm{\mathbf{a}}' \mathrm{\mathbf{x}} \ge b$ and $\mathrm{\mathbf{a}}' \mathrm{\mathbf{y}} \ge b$, respectively, and therefore belong to the same halfspace. Let $\lambda \in [0, 1]$. Then, $\mathrm{\mathbf{a}}'(\lambda \mathrm{\mathbf{x}} + (1 - \lambda)\mathrm{\mathbf{y}}) \ge \lambda b+ (1 - \lambda)b = b$, which proves that $\lambda \mathrm{\mathbf{x}} +(1 - \lambda) \mathrm{\mathbf{y}}$ also belongs to the same halfspace. Therefore a halfspace is convex. Since a polyhedron is the intersection of a finite number of halfspaces, the result follows from part (a).

The author provides a proof of this fact in his own question. However, after trying to understand this proof, a number of questions arose:

*

*The proof starts by supposing that $\mathrm{\mathbf{a}}' \mathrm{\mathbf{x}} \ge b$ and $\mathrm{\mathbf{a}}' \mathrm{\mathbf{y}} \ge b$. But, in the above definition of a polyhedron, it is said that $\mathcal{P} = \{ x \mid a^T_j x \le b_j, \ j = 1, \dots , m, \ c_j^T x = d_j, \ j = 1, \dots , p \}$; in other words, we have that $a^T_j x \le b_j, \ j = 1, \dots , m$, instead of $a^T_j x \ge b_j$, which it seems is what the proof has?


*The above definition also has the set condition $c_j^T x = d_j, \ j = 1, \dots , p$. Is this equivalent to $\mathrm{\mathbf{a}}'(\lambda \mathrm{\mathbf{x}} + (1 - \lambda)\mathrm{\mathbf{y}}) \ge \lambda b+ (1 - \lambda)b = b$ in the proof?


*The proof says that "Since a polyhedron is the intersection of a finite number of halfspaces, ...". What about the hyperplanes? I see no mention of them in the proof.
I would greatly appreciate it if people could please take the time to clarify these.

In answer to question 1, if you negate $a$ and $b$, you obtain $-a^Tx\le -b$.
For questions 2 and 3, a hyperplane is the intersection of the closed halfspaces it bounds.
