Let $V:=\{v_i\}_{i=1}^n$ be a collection of vectors in dimension $D$. Let $C$ be the convex hull (and its interior) of these vectors, including the origin (I think this is called the convex polytope?). So specifically, $C=\{u: u=\sum_{i=1}^n\lambda_iv_i, \lambda_i\geq 0,\sum_{i=1}^n\lambda_i\leq 1\}$.
Define the median vector $M$ to be the element-wise median of the vectors $V$. Are there any reasonable non-trivial conditions on $V$ would ensure that $M\in C$? Alternatively, what conditions on $V$ would ensure $M$ is not in $C$?
I was thinking that if you have a vector $c$ such that $\min_i(v_{ij})\leq c_j\leq\max_j(v_{ij})$, i.e. element-wise $c_i$ is in between the maximum and minimum values along each dimension, then $c$ is not necessarily an element of $C$. But I can't figure out how to nicely generalize this reasoning to the element-wise median.