### Symmetric functions and their extreme points

A symmetric function $f(x_1, \dots, x_n)$ is one that is invariant to permutations of $x$. In the 2D case, this means $f(a,b) = f(b,a)$. One basic example of a symmetric function is the area of a rectangle as a function of its side lengths, where $f(a,b) = a b$. It is well known that to maximize the area of a rectangle under a constraint on the perimeter, you should set all side lengths equal, to form a square. In this case, the maximum of this symmetric function occurs when $a=b$. While in general it is not always true that the extreme points of a symmetric function occurs when $a=b$, there are special cases when it does hold. For example, if $f(a,b)$ is a convex (concave) function, then a global minimum (maximum) will occur when $a=b$. Of course, the function above is not concave, but this property still holds. So the question arises: in what other situations does this nice property hold? In this blog post, I will state a general result that I found today regarding this pr