TITLE: Convex Hull of Path Connected sets
QUESTION [10 upvotes]: 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 $X$, $\operatorname{co}(X)$.
Is it enough to consider only pairwise convex combinations of points in $X$ to generate $\operatorname{co}(X)$? I.e.,
$$\left(\forall z\in \operatorname{co}\left(X\right) \right) \left(\exists\lambda \in \left[0,1\right] \land x_{0},x_{1}\in X \right) \left(z=\lambda x_{0}+(1-\lambda)x_{1} \right)$$
Also, if this is true, is it generalizable to more general topological spaces? Thanks!

REPLY [3 votes]: This is just a computational footnote to the thrust of this old question, but
I wanted to mention that finding the convex hull of a polygonal path is
computationally easier in $\mathbb{R}^2$ that finding the hull of unconnected
points: It can be computed in $O(n)$—linear time—compared to the $\Omega(n \log n)$ lower bound for unconnected points:

Melkman, Avraham A. "On-line construction of the convex hull of a simple polyline." Information Processing Letters 25, No. 1 (1987): 11-12.
  (ACM link.)
  
            
  

            
  
  Image from Joe Mitchell's course notes: 
  PDF download.