This is a pretty easy question to ask, but haven't seen it anywhere: suppose I have some continuous path $X$ in $\mathbb{R}^n$ and I want to get the convex hull of $X$, $co(X)$.

Question: Is it enough to consider only pairwise convex combinations of points in $X$ to generate $co(X)$? in other words:

$\forall z\in co\left(X\right)\exists\lambda\in\left[0,1\right]$ and $x_{0},x_{1}\in X$ such that $z=\lambda x_{0}+(1-\lambda)x_{1}$

Also: if this is true, is it generalizable to more general topological spaces?

Thanks!