Let $p_1,...,p_k$ be points in $\mathbb{R}^n$, let $S_1,...,S_m$ be subsets of $\{p_1,...,p_k\}$, and let $V_i$ be the convex hull of $S_i$. Is it true that \begin{equation} V:=\bigcup_{i=1}^m V_i \end{equation} is path-connected if and only if any two $p_i,p_j\in V$ can be joined in $V$ by a concatenation of line segments whose endpoints belong to $\{p_1,...,p_k\}$?
In other words, we can define a graph $G$ whose vertices are the points $p_i$ appearing in at least one $S_j$, and put an edge between $p_{i_1},p_{i_2}$ if and only if $\{p_{i_1},p_{i_2}\}\subseteq S_j$ for some $j$. Then my question is, is $V$ path-connected if and only if $G$ is a connected graph?
I feel like it should be easy to argue that yes but I have a hard time deducing something combinatoric from something topological.
This implication: "$\Leftarrow$" is obviously true.
But the other implication is false. Consider 4 points forming a square in $\mathbb{R}^2$:
$$p_1=(0,0),\ p_2=(0,1),\ p_3=(1,0),\ p_4=(1,1)$$
and let $S_1=\{p_1,p_4\}$ and $S_2=\{p_2,p_3\}$. Then $V$ is a union of two diagonals which is path connected. Note that $V_1\cap V_2=\{(1/2,1/2)\}$ which is not a vertex. And thus clearly $p_1$ and $p_2$ cannot be joined by concatenating line segments ending at any $(p_i,p_j)$. In this case such lines coincide with $V_1,V_2$.