For symmetric functions, people ask Do symmetric problems have symmetric solutions?, e.g., [3] and [4]. The answer is no in general. However, solutions of symmetric problems often exhibit certain symmetry. In [1], for a class of symmetric polynomials, the global minimum under some conditions is attained at some points with $|\{x_1, x_2, \cdots, x_n\} |\le 2$, i.e., at most two distinct components. In [2], for a linear combination of elementary symmetric polynomials, under some conditions, each of the local extrema ($n$-dimensional vector) has at most $k$ distinct components.

However, I am curious about whether there are examples (for differentiable symmetric functions under some conditions) in which all components of the global minimizer are distinct, if not impossible.

First, let us see an example. Under the conditions $x, y, z > 0$ and $xyz = 1$, the global minimum of $$g(x,y,z) = \frac{\sin \frac{\pi x}{2} }{x} + \frac{\sin \frac{\pi y}{2}}{y} + \frac{\sin \frac{\pi z}{2}}{z}$$ is attained at some points $(x_0, y_0, z_0)$ with exactly two of $x_0, y_0, z_0$ being equal, for example $(x_0, x_0, \frac{1}{x_0^2})$ where $x_0 \approx 2.852$ and the minimum $g_{\min} \approx 0.878$. Also $g(1,1,1) = 3 > g_{\min}$ and $\lim_{\min(x,y,z)\to 0^{+}} g(x,y,z) > 0.884 > g_{\min}$.

In the problem above, the components of the global minimizer is not distinct. Now, suppose $f: (0, \infty) \rightarrow \mathbb{R}$ is a differentiable function. Let $F(x, y, z) = f(x) + f(y) + f(z)$. I want to find some examples of $f$ such that under the conditions $x, y, z>0$ and $xyz=1$, the global minimum of $F(x,y,z)$ is attained at some point $(x_0, y_0, z_0)$ with none of $x_0, y_0, z_0$ being equal, if not impossible.

By the way, for cyclic symmetric functions, I found examples in which the global minimum is attained at some point with distinct components. For example, let $$F_1(a, b, c) = \frac{a^2b + 2a^2c + 2ab^2 + b^3 + 31abc}{(a+b+50c)(a+b+c)^2}$$ and let $G(a,b,c) = F_1(a, b, c) + F_1(b, c, a) + F_1(c, a, b)$. Then the minimum of $G(a, b, c)$ under the conditions $a, b, c \ge 0$ and $a+b+c=3$ is not achieved at $(1, 1, 1)$ or $abc=0$. Indeed, we have $G(1,1,1) = 37/156 \approx 0.2372$ and $$G(a, 3-a, 0) = \frac{49a^4-8094a^3+45900a^2-66177a-4050}{9(49a+3)(49a-150)} > 0.21, \quad \forall 0\le a\le 3.$$ However, $G(1/2, 1/8, 19/8) = 1018835/4907936 \approx 0.2076$; Actually, the global minimum is attained at some point with distinct components (also none of them is zero).

Reference:

[1] Vasile Cirtoaje, “The Equal Variable Method”, J. Inequal. Pure and Appl. Math., 8(1), 2007. Online: https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf

[2] Alexander Kovacec, et. al., “A note on extrema of linear combinations of elementary symmetric functions”, Linear and Multilinear Algebra, Volume 60, 2012 - Issue 2.

[3] R. F. Rinehart, "On Extrema of Functions which Satisfy Certain Symmetry Conditions", The American Mathematical Monthly, Vol. 47, No. 3 (Mar., 1940), pp. 145-152.

[4] William C. Waterhouse, “Do Symmetric Problems Have Symmetric Solutions?”, The American Mathematical Monthly, vol. 90, 1983, pp. 378-387.