are elementary symmetric polynomials concave on probability distributions? Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables.  Consider this polynomial as a function, in particular a function on probability distributions on $n$ items. It is not hard to see that this function is maximized at the uniform distribution.  I am wondering if there is a "convexity"-based approach to show this.  Specifically, is $S_{n,k}$ concave on probability distributions on $n$ items?

(I know this question is ancient, but I happened to run into it while looking for something else.)
While I am not sure if $S_{n,k}$ is concave on the probability simplex, you can prove the result you want and many other similar useful things using Schur concavity. A sketch follows.
A vector $y\in \mathbb{R}_+^n$ majorizes $x \in \mathbb{R}_+^n$ if the following inequalities are satisfied:
$$
\sum_{j=1}^i{x_{(j)}} \leq \sum_{j=1}^i{y_{(j)}}
$$
for all $i$, and $\sum_{i=1}^n x_i = \sum_{i=1}^n y_i$. Here $x_{(j)}$ is the $j$-th largest coordinate of $x$ and similarly for $y$. Let's write this $x \prec y$. For intuition it's useful to know that $x \prec y$ if and only if $x$ is in the convex hull of vectors you get by permuting the coordinates of $y$. 
A function is Schur-concave if $x \prec y \implies f(x) \geq f(y)$. A simple sufficient condition for Schur concavity is that $\partial f(x)/\partial x_i \ge \partial f(x)/\partial x_j$ whenever $x_i \le x_j$. It is easy to verify that $S_{n,k}$ satisfies this condition for any $n$,$k$.
Notice that $x=(1/n, \ldots, 1/n)$ is majorized by every vector $y$ in the probability simplex. You can see this for example by noticing that the sum of $i$ random coordinates of $y$ is $i/n$, so surely the sum of the $i$ largest coordinates is at least as much. Equivalently, $x$ is the average of all permutations of $y$. This observation, and the Schur concavity of $S_{n,k}$ imply $S_{n,k}(x) \ge S_{n,k}(y)$.
In fact, $S_{n,k}^{1/k}$ is concave on the positive orthant, and this implies what you want. This is itself a special case of much more powerful results about the concavity of mixed volumes. But the Schur concavity approach is elementary and pretty widely applicable.
