Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let $\overline{\mathrm{conv}}(C)$ denote the closure of the convex hull of a subset $C\subset\ell^2$.
Suppose $S$ is a non-empty, compact, convex subset of $\ell^2$, is it possible to write$S=\overline{\mathrm{conv}}\left(\bigcup_{n=1}^\infty[ S\cap(n\cdot H)]\right),$ where (for $n\in\mathbf N$ fixed) $n\cdot H=\{n\cdot x:x\in H\}$.
I think it is possible (since the Hilbert cube keeps getting 'thinner' in each coordinate), but I do not know how to prove it.