Given two convex function $f(x): R^2 \to R$ and $g(a, b) = f(a) + f(b): R^2 \times R^2 \to R$.
And a convex compact set $Z \subset R^2$.
We have $X^* = \{x^*\} = \text{argmax}_{x \in Z} f(x)$.
And $(A^*, B^*) = \{ (a^*, b^*) \} = \text{argmax}_{(a, b) \in Z \times Z; a+b \in Z} g(a, b)$.
Would $\cup (X^* \cap A^*, X^* \cap B^* ) != \emptyset $ hold ?