Carathéodory's theorem says that if a point $x$ of $R^d$ lies in the convex hull of a point set $P$, then there is a subset $P′ \subseteq P$ consisting of $d + 1$ or fewer points such that $x$ can be expressed as a convex combination of $P′$.
A recent result by Barman (see paper) shows an approximate version of the above theorem. More precisely, given a set of points $P$ in the $p$-unit ball with norm $p \in [2,\infty)$, then for every point $x$ in the convex hull of $P$ there exists an $\epsilon$-close point $x'$ (under the $p$-norm distance) that can be expressed as a convex combination of $O\left(\frac{p}{\epsilon^2} \right)$ many points of P.
Now, my question is that does the above result implies (or have some connection with) some kind of dimensionality reduction for the points in the convex hull of $P$. It seems intuitive to me (however I don't have a formal proof of it) - as for any point $x$ inside the $P$ there is a point (say) $x'$ in a close neighborhood of $x$ which can be written as convex combination of constant many points of $P$, which in some sense dimensionality reduction of $x'$.
Pls let me know if I am able put my question clearly.
Thanks.