Consider a matrix $A$ of size $(n, m)$ and its LU-decomposition with partial pivoting:
$$PA =LU$$
If we look at the $A$ as set of $n$ $m$-dimensional points, then first $m$ points in $PA$ would be linearly independent, if and only if $\text{det} U \neq 0$, because $\text{rank} U = \text{rank} A$ and $\text{rank} L \geqslant m$ by definition.
Let $A'$ be first $m$ points in $PA$.
Now consider a convex hull of point set $A$; and my question is:
Is $A'$ subset of $CH(A)$?
It might be implementation issue, I am using function scipy.linalg.lu