Assume $S_1+S_2+\dots+S_n=S$ is a constant. Also, assume $S_i$'s are upper bounded by $U$ where $S/n < U < 2S/n$ and lower bounded by $L$ where $0 < L < S/n$. This is obvious that the maximum of $S_1 S_2 \dots S_n$ is $(S/n)^n$. But, what is the minimum of $S_1 S_2 \dots S_n$ considering the upper bound $U$ for $S_i$'s?
Note: For $n = 2$, it is obvious that the minimum is $U(S-U)$. How about $n = 3$ or more?