Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$. Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable such that $\Pr[X_i = 1] = x_i$ and $\Pr[X_i = 0] = 1 - x_i$. All $X_i$'s are independent, and let $X = X_1 + \dots + X_n$. Let $f : P \to \mathbb{R}$ be defined such that $f(x) = \mathbb{E}[\min(X, k)]$.

Is this function convex? Is there anything known about this function?

We proved that $f(x)$ is minimized when $x = (\frac{k}{n}, \dots, \frac{k}{n})$ for any $k \leq n$, and numerically verified $f(x)$ is convex for small values of $n$ and $k$.