We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant in $(0,1)$).
Question: How can we prove or disprove that, for all convex $C\subseteq [0,1]^d$ and all $d\in\mathbb{N}$, there exists a set of $n$ points lying on the boundary of $C$ where $n$ grows at most polynomially in $d$, such that the volume of their convex hull $C'$ is lower bounded by a constant independent of $d$?
Note: The main intuition underlying this conjecture is the idea that (very) informally, for $d\gg 1$ if there are areas where the boundary of $C$ is far from being linear (e.g., portions of $(d-1)$-balls), the volume of the regions of $C$ bounded by such areas is disregardable compared to the volume of $C$. More generally, the same holds for areas formed by $\omega(\mathrm{poly}(d))$-many linear regions that can be viewed as good approximations of manifolds with significant curvature.