Meaning cone, ray, fan for polytopes I'm trying to understand some mathematical operations and definitions for my project. Could you explain the meanings? $P_b =\{x∈\mathbb{R}^d : Ax≤b\}$
is a polytope. Let's have a 10x3 matrix $Ax≤b$
$$
\begin{bmatrix}
0 & 1 & 1 \\ 
1 & 0 &1 \\ 
1 & 0 & 0\\ 
0 & 1 & 0\\ 
0 & 0 & -1 \\ 
0 &  0& 1\\ 
0 &  -1& 0\\ 
0 &  -1& 1\\ 
-1 &  0& 0\\ 
-1 &  0& 1
\end{bmatrix}
\begin{bmatrix}
x_1\\ 
x_2\\ 
x_3
\end{bmatrix}
\leq
\begin{bmatrix}
4\\
4\\
3\\ 
3\\ 
0\\
2\\
0\\
1\\
0\\
1 
\end{bmatrix}
$$
Let us denote the set of such right hand side vectors $b$ by
$U(A)=\{b∈\mathbb{R}^m : P_b \neq ∅\}.$

Switching from the H-representation to its V- representation, the cone U (A) is generated by 9 rays and 3 lines in $Z^{10}$.
The normal cone of a face F of a polytope P in $R^d$ is the set
$N(F;P)=\{v∈\mathbb{R}^dd : v^⊤x=h(P,v)~for~all~ x∈F\}.$
The dimension of the normal cone of a $k$-dimensional face is $(d − k)$. The normal fan $N(P)$ of $P$, which is the collection of the normal cones of all faces of $P$ , is a complete fan in $\mathbb{R}^d$.
I have difficulty in understanding of the terms.
So, what is the exact meaning of $U(A)$ here? What are the $b_{1,2,...}$ letters and how obtained? How is the 9 rays and 3 lines expression written? Can you explain also meanings of normal fan,cone and complete fan?
Article can be found from here. I want to grasp all of the article also I have to, but at least up to there is enough.
@Edit:
I think it's a kinda Millenium Problem. If not, isn't there any helper mathematician?
 A: The article investigates the set $U(A) = \{b : \exists x\; Ax \leq b \}$. In words, that is the set of vectors $b$ for which $Ax \leq b$ has a solution, that is, for which the set $\{x : Ax \leq b\}$ is not empty. Via Farkas Lemma, they express the set as $U(A) = \{ b : y^Tb \geq 0 \; \forall y \in S\}$ with $S = \{ y \geq 0 : A^Ty=0 \}$.
Without going into how the set $S$ is derived, we can already make a few remarks. Since $A \in \mathbb{R}^{10 \times 3}$ in the example, $S \subset \mathbb{R}_+^{10}$. Moreover, if $y \in S$, then also $\alpha y \in S$ for any $\alpha \geq 0$. Therefore, $S$ is a cone. The set $S$ is also convex, so it is a convex cone. The set $\{ \alpha y : \alpha \geq 0 \}$ is called a ray, which is just a half-line with one endpoint at the origin. A ray of $S$ that cannot be expressed as a convex combination of other rays of $S$ is called an extreme ray of $S$. Figure 1 of this article has an illustration of the extreme rays of a cone. If $-y\in S$ for some $y\in S$, $S$ is said to contain a line. A cone that does not contain any lines is a pointed cone. A cone is defined (or generated) by the set of its (possibly infinite number of) lines and rays.
To verify if a given vector $b$ is an element of $U(A) = \{ b : y^Tb \geq 0 \; \forall y \in S\}$ one does not need to check all elements in $S$, but it suffices to check whether $y^Tb \geq 0$ for one nonzero element of each extreme ray of $S$ and $y^Tb=0$ for one nonzero element of each extreme line of $S$ (it then automatically holds for all elements in $S$). The remainder of the article is about deriving the extreme rays of $S$. As a spoiler they already provide the final result (those 9 inequalities involving $b_i$), but you cannot understand why they are correct without reading the rest of the paper. At this point you can however check that indeed $A^Ty=0$ for each vector $y$.
The set $S$ is a polytope, since it is the finite intersection of closed halfspaces. The face of a polytope is visually one of the sides, edges or corners of the polytope. The mathematical definition of a face is $F = S \cap\{y:c^Ty=d\}$ where $c$ and $d$ are such that $c^Ty \leq d$ for all $y \in S$ ($c^Ty \leq d$ is a valid inequality). Each face $F$ has an associated normal cone, $N(F,S) = \{ v : v^Ty \geq v^Tz \; \forall y \in F \; \forall z \in S\}$. The normal cone (which is a convex cone) is the set of all vectors that are orthogonal to the face. An illustration can be found in this question (left panel). The set of all normal cones is called the normal fan. An illutration is found here, and indeed it covers every angle which means it is a complete fan.
