For symmetric functions, people ask *Do symmetric problems have symmetric solutions?*, for example, (3) and (4).

The answer is not in general. However, symmetric problem solutions often exhibit some symmetry.

In (1), for a class of symmetric polynomials, the global minimum in some conditions

it is reached at some points with $ | {x_1, x_2, cdots, x_n } | le 2 $, that is, at most two different components.

In (2), for a linear combination of elementary symmetric polynomials, in some conditions, each of the local extremes ($ n $-dimensional vector)

has at most $ k $ Different components

However, I'm curious to know if there are examples (for **differentiable** symmetric functions in some conditions) in which all components of the global minimizer are **different**, if not impossible.

First, let's see an example. Under the conditions $ x, y, z> 0 $ Y $ xyz = 1 $,

the global minimum of $$ g (x, y, z) = frac { sin frac { pi x} {2}} {x} + frac { sin frac { pi y} {2}} {y} + frac { sin frac { pi z} {2}} {z} $$

it is reached in some points $ (x_0, y_0, z_0) $ with exactly two of $ x_0, y_0, z_0 $ being the same

for example $ (x_0, x_0, frac {1} {x_0 ^ 2}) $ where $ x_0 approximately 2,852 $ and the minimum $ g min approx. $ 0.878.

as well $ g (1,1,1) = 3> g min Y $ lim _ { min (x, y, z) a 0 + g (x, y, z)> 0.884> g {{min} $.

In the previous problem, the components of the global minimizer are no different.

Now suppose $ f: (0, infty) rightarrow mathbb {R} $ It is a differentiable function.

Leave $ 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 $ Y $ xyz = 1 $,

the global minimum of $ F (x, y, z) $ is reached at some point $ (x_0, y_0, z_0) $

with none of $ x_0, y_0, z_0 $ being the same, if not impossible.

By the way, for cyclic symmetric functions, I found examples where the global minimum

it is reached at some point with different components. For example,

leave

$$ F_1 (a, b, c) = frac {a ^ 2b + 2a ^ 2c + 2ab ^ 2 + b ^ 3 + 31abc} {(a + b + 50c) (a + b + c) ^ 2} $$

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 $ Y $ a + b + c = $ 3 it is not achieved in $ (1, 1, 1) $ or $ abc = 0 $.

In fact, we have $ G (1,1,1) = 37/156 approximately 0.2372 $ Y

$$ G (a, 3-a, 0) = frac {49a ^ 4-8094a ^ 3 + 45900a ^ 2-66177a-4050} {9 (49a + 3) (49a-150)}> 0.21, forall 0 le a le 3. $$

But nevertheless, $ G (1/2, 1/8, 19/8) = 1018835/4907936 approximately 0.2076 $;

In reality, the global minimum is reached at some point with different components (neither is zero).

Reference:

(1) Vasile Cirtoaje, "The same variable method", J. Inequal. Pure and applied. Mathematics, 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 extremes of linear combinations of elementary symmetric functions",

Linear and multilinear algebra, Volume 60, 2012 – Number 2.

(3) R. F. Rinehart, "On the extreme functions that fulfill certain symmetry conditions",

The American Mathematical Monthly, vol. 47, no. 3 (March 1940), p. 145-152.

(4) William C. Waterhouse, "Do symmetric problems have symmetric solutions?", The American Mathematical Monthly, vol. 90, 1983, p. 378-387.