IX OBM

Suppose $n$ is $3m$. Take the points to be $$\begin{gathered} (1,0,0),(2,0,0),\dots ,(m,0,0), \\ (0,1,0),(0,2,0),\dots ,(0,m,0), \\ (0,0,1),(0,0,2),\dots ,(0,0,m) \end{gathered}$$ Take $P$ to be $(x,y,z)$. Then the sum is $2m|x|+|x-1|+|x-2|+\cdots +|x-m|$ plus similar terms in $y$ and $z$. Evidently we can minimize for $x,y,z$ separately. Looking at the terms in $x$, it is clear that if we increase $|x|$ by $k>0$, then we increase $2m|x|$ by $2mk$. At most we can reduce the other terms by $mk$, so we increase the sum by at least $mk$. Similarly, if we reduce $|x|$ by $k>0$, we reduce the sum by at least $mk$. So to minimize the sum we must take $|x|=0$. Similarly for $y$ and $z$. Hence $P$ is at the origin which is outside the convex hull. If $n$ is $3m-1$ or $3m-2$, then we drop $(0,0,m)$ and/or $(0,m,0)$. Then the argument is strengthened for the $y$ and $z$ coordinates. For the $x$ coordinate we have at worst $(2m-2)|x|+|x|+|x-1|+\cdots +|x-m|$. If $n>6$, then $m>2$, then $2m-2>m$ and the same argument works. If $n=5$, then we have $3|x|+|x|+|x-1|$ for the $x$-coordinate, and the same argument works.