Optimum (min/max) of a symmetric function

[NaturePaper]

Hi all,
I'm wondering if the following argument is right:

"The optimum (minimum/maximum) value of a symmetric function f(x_1,x_2,...,x_n) (By 'symmetric' I mean that f remains same if we alter any x_i's with x_j's), if exists, should be at the point x_1=x_2=...=x_n".

Please help me by proving (or disproving, i.e, a counter example) this argument. I have examine some well-known symmetric functions and these obey the rules. However, a general proof (or reference to literature, document etc.) is needed.


[I think this may be right, in the sense that the graph of the function (in n+1 dimensional space) should be symmetric about all axes, so if f has optimum at x_1=a, by the symmetry of the curve, x_2=...=x_n=a.]


Regards,
NaturePaper

 

[yyat]

I assume that you mean that the optimum, if it exists, should be unique. Say it is some point $$(a_1,\ldots,a_n)$$, but then $$a_1=\ldots=a_n$$, because otherwise one could interchange two coordinates to obtain another optimum.

 

[NaturePaper]

@yyat,
Oh..., Let us consider the global optimum (min/max) for f. For convex f, it is known that every local optimum is the global optimum. However, I don't know whether f is convex or not. But I think the following discussion is right:

Let f has a global optimum (by definition, it is unique). Then the set of points where f attains it should contain the point x_1=x_2=...=x_n= some constant, by your argument.

Am I right?

Regards
NaturePaper

 

[yyat]

By uniqueness I meant that the optimum is only achieved at a single point, otherwise the argument becomes false. For example, consider the function on R^2 that is mostly 0 but has two http://en.wikipedia.org/wiki/Bump_function" at (1,0) and (0,1) that are mirror images of themselves with respect to the diagonal (x_1=x_2). Then the function has two global maxima, but none of them on the diagonal.

 

[NaturePaper]

Anyway, if I restrict f to be symmetric, even function [i.e. f(-X)=f(X)], any number of time differentiable, then is it possible to make any comment about its global optimum?
At least, is f convex etc...?

Actually, my function f is like f=[g(x_1,x-2,...,x_n)]^2, where g is symmetric, bounded and any time differentiable. Is it possible to comment about the convexity of f etc?

Regards,
NaturePaper