Let $T_1$ and $T_2$ be two regular $(n-1)$-dimensional simplices with vertices $$(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t),$$ and $$(t-n+1,1,\ldots, 1), (1, t-n+1, \ldots, 1), \ldots, (1,1, \ldots, t-n+1),$$ respectively. Suppose also $m < t < m+1$ for some integer $m$, $0 < m < n$. The intersection of these simplices is a polytope $P$.
As was shown in this answer, the vertices of $P$ are the $n {{n-1} \choose m}$ points that have $m$ coordinates $1$, $n-m-1$ coordinates $0$ and one $t - m$.
How to show that the point of the polytope that maximizes the minimum distance to a vertex is a barycentre.
Thank you.