Let $-N\leq t \leq N$.
Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, \ldots, 1)\ldots (1,1, \ldots, t-N+1)$.
The intersection $A \cap B=P(N, t)$ is a polytope with $N {N-1 \choose m}$ vertices, where $m<t<m+1, \, 0<m<N, \, m \in N$. These $N {N-1 \choose m}$ vertices have $m$ coordinates $1$, $N-m-1$ coordinates $0$ and $1$ coordinate $t-m$.
Polytope polytope $P(N,t)$ can be subdivide into $N {N-1 \choose m}$ equal parts,$Q_v$, such that each part would correspond to one vertex of $P(N,t)$. (see Splitting polytope into equal parts)
Let $f$ be some function defined on $P(N,t)$.
Question: Where the maximum and the minimum values of $f$ on $Q_v$ would be attained?