Let $\gamma :[0,1] \mapsto \mathbb{R}^{n}$ be a curve such that all the leading principal minors of the matrix $(\gamma^{(1)}(t), \ldots, \gamma^{(n)}(t))$ are nonvanishing as $t \in [0,1]$.
Theorem: the minimal number $k$ points required to represent any interior point of $\mathrm{Conv}(\gamma([0,1]))$ as a convex combination of $k$ points of $\gamma([0,1])$ equals $\lfloor \frac{n}{2}\rfloor+1$.