We work over $\mathbb{R}^N$. Let $V$ be the corners of the unit cube $[0,1]^N$, or equivalently the set of vectors whose coordinates take values $0$ or $1$.
Let $d:\{0, \ldots, N\} \to \mathbb{R}_+$ be a function such that $nd(n)$ is concave, with maximum attained at $n_{\text{max}}$.
Define $S = \{d(\|v\|_1)v: v\in V\}$. Let $C$ be its convex hull. I want to show that if $v\in V$ and $\|v\|_1 > n_{\text{max}}$, $v$ is not an extreme point of $C$.