Description of functionals that are minimized by $\Phi$-triangulations

We obtain condition for a function $f$ defined on the set of simplexes $S$ under which the values $F(T)=\sum\limits_{S\in T}f(S)$ or $F_f^m(T)=\max\limits_{S\in T}f(S)$ are minimal for $\Phi$-triangulations of $T$. As consequences, we also obtain certain extremal properties of the classical Delaunay triangulation.