Making sense of the definition of Polyhedron A polyhedron is defined as the solution set of a finite number of linear equalities
and inequalities:
$$P = \{x | a^T_j x ≤ b_j , j = 1, . . . , m, c^T_j x = d_j , j = 1, . . . , p\}$$
A polyhedron is thus the intersection of a finite number of halfspaces and hyper-
planes. Affine sets (e.g., subspaces, hyperplanes, lines), rays, line segments, and
halfspaces are all polyhedra.
I understand the use of inequalities in the equation. It is used to select the halfspaces included by the polyhedron and thus form a closed structure. But I don't understand:


*

*What is the role of the equalities $c^T_j x = d_j , j = 1, . . . , p$

*And my second question is how is the 'intersection of a finite number of halfspaces and hyper-planes' a polyhedron? i.e for example if we take 2 halfspaces $a^T_1 x ≤ b_1$ and $a^T_2 x \geq b_2$ it will form an unending triangle (informally speaking) covering a sector of the plane. How is this a polyhedron?
NOTE: I am not a mathematician, so use of intuitive terms rather than technical will be more helpful to me.
 A: For your first question, each equality $c_j^T x = d_j$ means that the polyhedron will live inside a particular hyperplane. Intuitively, intersections of multiple hyperplanes can be seen as reducing the dimension of your polyhedron. For instance if $P \subseteq \mathbb{R}^2$ is a $2$-dimensional polyhedron living in the $xy$-plane and we intesect this with the hyperplane $x = 0$ then we are looking at the intersection of $P$ with the $x$-axis. This could be empty, a point or an interval.
For your second question, your example is certainly a polyhedron however not a bounded one. Intuitively the definition of polyhedron is describing which solid objects have 'flat' bounding faces. Usually the important things for a polyhedron are knowing which points lie inside and what its faces look like. Generally it is not a problem if the polyhedron is not bounded.
A: For the first question, a polyhedron is the intersection of a finite number of half-spaces and hyper-planes. Only hyperplanes can't create a polyhedron. In that case, $C_{j}^{T} x=d_{j}$ is not enough to create the polyhedron.
Polyhedron can be unbounded.
