I found this very interesting paper online which yields a version of Caratheodory's Theorem independently of its dimension (Theorem 1.1 therein), i.e.
Let $P$ be a set of $n$ points in $R^d$, $r ∈ [n]$ and $a ∈ conv P$ . Then there exists a subset $Q$ of $P$ with $|Q| = r$ such that $d(a, conv Q) < \frac{diam P}{\sqrt{2r}}$.
I would love to understand how they derive this bound, but I have troubles finding those arguments withing the paper. There is no proof of it. Maybe it is a well-known result from some other papers? Or it follows from some of the later results?