Question
Assume that $K\subseteq \mathbb{R}^n$ is a convex set. For some $m\geq n+1$, let $C_1,…,C_m$ be convex sets with the property that the intersection of every $n+1$ of them contains a translated copy of $K$. Prove that $\bigcap_{i=1}^m C_i$ contains a translated copy of $K$.(A translated copy of $K$ is a set $x+K$, where $x\in \mathbb{R}^n$).
Attempt
We use prove by induction. For $m=n+1$, we see that $\bigcap_{i=1}^m C_i$ contains a translated copy of $K$. Therefore, the statement holds trivially.
Suppose the statement holds for $m=k$, where $k\geq n+1$, we want to show that the statement holds for $m=k+1$. Consider the set $C_1,…,C_k,C_{k+1}$. Since $\bigcap_{i=1}^k C_i$ is non-empty (it contains $K$ by inductive hypothesis), then by Helly’s theorem, $\bigcap_{i=1}^{k+1} C_i$ is non-empty.
My challenge I want to make a case that $\bigcap_{i=1}^{k+1} C_i$ has a translated copy, and since $\bigcap_{i=1}^{k+1} C_i$ is non-empty, it has an element $x$. So, the translated copy is $K+x$. However, I don’t know how to go about this. I believe there should be a way to link my thoughts logically and convincingly.
Your help would be greatly appreciated.