Minimizing polynomial function over the standard simplex

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 $$x_1 = x_2 = \frac 12$$.

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