Exploiting symmetry to prove results

Question

Given a real number number $n$, find a partition whose product is least / max. I.e: $n=a_1+a_2+a_3+ \cdots+a_m$ (Here $m$ is a variable as well)

What can the maximum/minimum value of $a_1a_2a_3\cdots a_m$ be?

I am attempting to use a symmetry argument to show that $a_i=a_j \forall (i,j) \in {1,2,\cdots,m}$

I am trying to prove this condition by means of contradiction.

Suppose $\exists k,l$ s.t $a_k \not = a_l$ but the product is max/min. Then either $a_k > a_i$ or $a_k < a_i$. Consider the first case; since multiplication is commutative, hence we can swap $a_i$ and $a_k$ within the product. This would show that $a_k < a_i$. Hence, it contradicts our assumption. By similar reasoning, $a_k<a_l$ can also be shown to be false. Hence, $a_I=a_j \forall (i,j) \in {1,2,3, \cdots m}$

Is my reasoning correct ? If so, Is it a well known technique? I want to see further applications of this technique in problem solving.

If I have made a mistake, please highlight it.

Answer 1

Suppose $m=3$, $a_1 = n$, $a_2 = K$, $a_3 = -K$ with arbitrary $K$. Then the product is $-a_1 K^2$, which is not bounded. Hence at least the minimum or the maximum fails to exist.

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Your proof has a problem: It assumes that solutions must be unique, or otherwise it does not use the fact that the solution is a maximum or a minimum at all.

With your technique, try starting from $3 = 1 \cdot 3$, then also $3 = 3 \cdot 1$, and reach a contradiction.

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You can find local minimums or maximums for the function, and so necessary conditions for your problem, if they exist, by using analysis: optimize the value of the product and restrict the variables to the manifold defined by the sum condition.

Answer 2

Without the assumption that all $a_i \geq 0$, the product can attain any value once $m \geq 3$.

Assuming positivity, yes, there are symmetry arguments similar to what you describe and they are used to prove the Arithmetic Mean - Geometric Mean inequality, which is a formal statement of the principle that the product is maximized at the most symmetric solution.

For the minimum, the product can be $0$ if $a_i$ are allowed to equal $0$ and can be arbitrarily close to $0$ if the constraint is $a_i > 0$.