Does Approx Carathéodory's theorem implies dimensionality reduction
The approximate Caratheodory theorem goes back to the 60s, and probably way earlier than that (it follows for example from the mistake bound of the preceptron algorithm analysis). As for the dimensionality reduction, the answer is no - the supporting subsets are different subsets, and their number is too large. In particular, the number of possible subsets is $n^{O(1/\epsilon^2)}$ - but there is some connection... 

Specifically, let $P$ be a set of $n$ points in high dimenionsional Euclidean space of diameter $1$. Let $Q$ be an $\epsilon$-net in the convex-hull of $P$. That is, any pair of points of $Q$ is in distance at least $\epsilon$ from each other, and every point of $CH(P)$ is in distance at most $\epsilon$ from some point of $Q$. 

Now, by the approximate Caratheodory theorem, we know that $|Q| = n^{O(1/\epsilon^2)}$. Now, imagine that you do some experiment, and with probability half the experiment succeeds for half the points of $Q$ (or, more formally, half the pairs of $Q \times Q$ -- since we look on vectors formed by differences of points of $Q$). How many times do you have to repeat the experiment till all the points are served? Well, roughly $\log_2 |Q| = O(\epsilon^{-2} \log n)$, which is, surprise surprise, the target dimension in the JL lemma. This is of course, does not imply the JL lemma - it is somewhat of a "coincidence" - a wrong calculation that gives the right bounds...

There is a useful lesson here however - a set of $n$ points in high dimensions, induces roughly $n^{O(1/\epsilon^2)}$ points that are $\epsilon$-distinct, independent of the ambient dimension.