Max min problem with sum constraint

Question

The problem is to maximize the minimum $x_i,\:i=1,\ldots,n$, such that all $x_i\geq 0$ and, $$\sum_{i=1}^n x_i\leq t$$ where $t>0$ is a constant. I think the solution is to have equal $x_i$s, i.e. $x_i=\frac{t}{n}$, but I don't know how to prove this.

Answer 1

Let $m=\min_i x_i$. Then, $$ t=\sum_ix_i\geq\sum_im=nm\implies m\leq\frac{t}{n}. $$ It remains to demonstrate that $\frac{t}{n}$ is achievable. But this is easy: take $x_1=\cdots=x_n=\frac{t}{n}$. So the maximum of the minimal $x_i$ is $\frac{t}{n}$.

Answer 2

Your answer is correct. For a proof, suppose wlog $x_1 \le x_2 \le \ldots \le x_n$ so $x_1 = \min_j x_j$. Define $a_j := x_j - x_1 \ge 0$ for all $j$. The constraint becomes $$\sum_j x_j = nx_1 + \sum_j a_j \le t$$ where $\sum_j a_j \ge 0$. The objective function reads $$\max x_1 = \max \frac{t - \sum_j a_j}{n}$$ This attains the maximum for $\sum_j a_j = 0$, when $x_j=x_1= t/n$.