Finding whether $n$ polytopes have nontrivial intersection from pairwise comparisons I have a set of $n$ convex polytopes of the form 

$$\mathcal{L_i} = \{ \beta \mid C_i \beta \leq 0 \}$$

where $C$ is a matrix and $\beta$ is a vector. I know that for each pair of polytopes $$(\mathcal{L}_k, \mathcal{L}_j)$$ there is a point of intersection other than the trivial solution $\beta = 0$. Additionally, I know that $C_i$ is such that each entry of $\beta$ must be greater than or equal to zero. What I want to know is whether this implies that there exists a nontrivial point of intersection of all $n$ polytopes, or if there is some counterexample? 

Thanks!
Your question is related to how to represent the geometric intersection objects. Pairwise comparisons can get a geometric intersection graph, however, the problem is that two objects are possibly not close to each other in this intersection graph, even originally their geometric distance is very short. So it is easy to find counterexamples for the specific question you asked: whether this implies that there exists a nontrivial point of intersection of all $n$ polytopes. 

Given a geometric intersection graph, deciding which graph class it belongs to, the so-called recognition problem is not easy: Shaefer [1] proved that for intersection graphs of convex sets, the recognition problem is  $\exists \mathbb{R}$-complete(It lies between NP and PSPACE). 

[1] M. Schaefer. Complexity of some geometric and topological problems. In Proc. 17th Int. Symp. Graph Drawing (GD), pages 334–344, 2009.