On a pair of simultaneous functional equations

For each p in the simplex P of $\text{R}$^k we introduce convex subsets of P, Π_I(p) and Π_II(p). For f a real function on P we define Cav₁f to be the smallest function greater than f on P and concave on Π₁(p) for each p in P (and similarly Vex_IIf). Given u a continuous real function on P we prove that the following problems:

$\text{Minimize}\text{f;f:→R, f?}\text{Cav}{}_{\mathrm{I}}\text{Vex}{}_{\mathrm{II}}\text{max}\text{{u,f}}$

$\text{Minimize}\text{f;f:→R, f?}\text{Vex}{}_{\mathrm{II}}\text{Cav}{}_{\mathrm{I}}\text{min}\text{{u,f}}$

have the same solution which is also the only solution of f = Vex₁₁ max{u,f} = Cav₁ min{u,f}. This is an extension of a former proof by Mertens and Zamir for the case where P is a. product of convex R and S with Π_I(p) = r × S and Π_II(p) = R × s.