I have the following optimization problem
$$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{i=1}^k x_i \prod_{j=1}^{i-1} (1-x_j)\\ \text{subject to} & \displaystyle\sum_{i=1}^k x_i = 1\\ & x_i \in [0,1], \quad \forall i\end{array}$$
whose solution I think is easy to infer. Is it true that the minimizer is $x_i = \frac 1k$ for all $i \in [k]$? If so, how to prove it?
For example for $k=2$, the minimizer seems to be $x_1 = x_2 = \frac 12$.