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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 $x_1 = x_2 = \frac 12$.

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  • $\begingroup$ Have you tried for $k=3$ and $k=4$? $\endgroup$ – Rodrigo de Azevedo Apr 4 at 19:54

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