I have a function that is defined on only positive orthant $\mathbb{R}^3_{+}$. The function is monotonic increasing along any of three axis.
(This is the same monotone vector function as defined on page 108 in Convex Optimization by Boyd here.)
Question:
If there exists a symmetry as well in the above function; in two of its arguments then: my intuition suggests that such a monotone increasing function which is symmetric too (in its two variables, let's say $X_1$ and $X_2$) would always have a maxima on the plane that bisects the whole 3D region into two: such that it passes through (contains) the third axis ($X_3$) with first and second co-ordinates to be the same. (i-e parametric equation of a maxima containing plane should look like this $(X_1,X_1,X_3):=(\alpha,\alpha, x_3)$ where $0 \le \alpha \le \infty $ and $0 \le x3 \le \infty $.)
If this is true? How can I justify it mathematically?