This is a question from the book lectures on Discrete Geometry:
Let $C_1, \dots, C_n$ be convex sets in the plane such that each 4-tuple of them contains a ray in the intersection. Prove that $\bigcap_{i=1}^n C_i contains a ray. My (wrong I think) idea how to solve it is the following:
We proceed by induction on $n\ge 4$. Base case is obvious so we will show that $n\implies n+1$. Let $a_i + tr_i, t\ge0$ be the ray contained in $\bigcap_{j \neq i}C_j$. This gives us a set of at least 5 vectors in $\mathbb{R}^4$ $\{(a_i, r_i): i\ge 0\}$. These vectors are linearly independent so we can split $[n]$ into disjoint sets $A,B$ such that $\sum_{i\in A}a_i \lambda_i = \sum_{j \in B}a_j\mu_j$ and similarly for $r_i$
From here my idea was to show that ray $\sum_{i\in A}a_i \lambda_i + t\sum_{i\in A}r_i \lambda_i$ will be in the intersection but I am not sure if this works, particularly if the sums above are zero which can happen it shouldn't work. Thus I am interested in if my solution can be fixed or finding alternate solutions to this problem.