Let:
- $f(x,y)$ - a strictly-concave function, monotonically increasing in $x$ and $y$;
- $A,B$ - two compact and convex sets in the positive quadrant;
- $C$ - their Minkowski sum, $A+B$;
- $(x_A,y_A)$ - the point in $A$ that maximizes $f$ (it is unique since $f$ is strictly-concave);
- $(x_C,y_C)$ - the point in $C$ that maximizes $f$.
I am trying to prove (or find a reference to) the following conjecture: $$x_C\geq x_A, \,\,\,\,\, y_C\geq y_A$$ I.e, if we Minkowski-add to $A$ a positive set ($B$), then the maximum point of $f$ moves only rightwards and/or upwards.
Is this conjecture true?
Maybe it is true if we assume that $f$ is symmetric, i.e, $f(x,y)=f(y,x)$?