Intersection of polytopes Given two polytopes, $\mathcal{S}$ and $\mathcal{U}$, they can be described as the intersection of finitely many halfspaces, i.e., $\mathcal{S} := \{x \in \mathbb{R}^n \mid A x \leq b \}$ and $\mathcal{U} := \{ x \in \mathbb{R}^n \mid C x \leq d \}$, where $A \in \mathbb{R}^{p \times n}$, $C \in \mathbb{R}^{q \times n}$ have full column rank, $b \in \mathbb{R}^p$ and $d \in \mathbb{R}^q$.
Is anyone aware of conditions on $A$, $C$, $b$ and $d$ guaranteeing the nonemptiness of the intersection set, i.e., $\mathcal{S} \cap \mathcal{U} \neq \emptyset$?
 A: Let $m$ and $n$ be natural numbers, $k=1,\dots, m$,  $a_k=(\alpha_{k1},\alpha_{k2},\dots, \alpha_{kn})\in\Bbb R^n$, and $\beta_k\in\Bbb R$. Given a system  $$(\alpha_k, x)\le \beta_k\,  \label{1}\tag{1}$$ of linear inequalities, there is a general problem to find conditions of its compatibility, that is when has at least one solution. In [Cha], beginning from §37 are listed most known compatibility conditions.
Theorem 1. The system \eqref{1} is compatible iff an equality $\sum_{k=1}^m \lambda_k\alpha_k=0$ with non-negative scalars $\lambda_k$ implies an inequality $\sum_{k=1}^m \lambda_k\beta_k \geq 0$.
Theorem 3. (S. N. Chernikov). The system \eqref{1} of rank $r$ is compatible iff the matrix of the system has a minor of order $r$
$$\delta=\left|\begin{matrix} 
\alpha_{i_1j_1} & \alpha_{i_1j_2} & \dots & \alpha_{i_1j_r} \\
\alpha_{i_2j_1} & \alpha_{i_2j_2} & \dots & \alpha_{i_2j_r} \\
\dots & \dots & \dots & \dots \\
\alpha_{i_rj_1} & \alpha_{i_rj_2} & \dots & \alpha_{i_rj_r} \\
\end{matrix}
\right|\ne 0,$$
such that
$$\frac 1{\delta}\left|\begin{matrix} 
\alpha_{i_1j_1} & \alpha_{i_1j_2} & \dots & \alpha_{i_1j_r} & \beta_{i_1}\\
\alpha_{i_2j_1} & \alpha_{i_2j_2} & \dots & \alpha_{i_2j_r} & \beta_{i_2}\\
\dots & \dots & \dots & \dots & \dots \\
\alpha_{i_rj_1} & \alpha_{i_rj_2} & \dots & \alpha_{i_rj_r} & \beta_{i_r}\\
\alpha_{ij_1} & \alpha_{ij_2} & \dots & \alpha_{ij_r} & \beta_{i_r}\\
\end{matrix}
\right|\ge 0,$$
for all $i\ne i_1, i_2,\dots, i_r$.
Theorem 2. (§38). The system \eqref{1} is not compatible iff its matrix contains $p$ rows, containing a $p\times (p-1)$-matrix
$$\underline{A}=
\begin{pmatrix} 
\alpha_{i_1j_1} & \alpha_{i_1j_2} & \dots & \alpha_{i_1j_{p-1}} \\
\alpha_{i_2j_1} & \alpha_{i_2j_2} & \dots & \alpha_{i_2j_{p-1}} \\
\dots & \dots & \dots & \dots \\
\alpha_{i_pj_1} & \alpha_{i_pj_2} & \dots & \alpha_{i_pj_{p-1}} \\
\end{pmatrix}$$
such that $\delta_i\delta_{i+1}<0$ for $i=1,\dots, p-1$ and $\sum_{i=1}^p \beta_{k_i}|\delta_i|<0$ [I guess here should be $\beta_{i}$ instead of $\beta_{k_i}$. AR], where $\delta_i$ is a minor of order $p-1$, obtained from $\underline{A}$ by removing its $i$-th row.
References
[Cha] V. S. Charin, Linear transformations and convex sets, Kyiv: Vyshcha shkola, 1978, in Russian.
