Let $\mathbf{p}=(p\_1,\dots,p\_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X\_1,\dots,X\_m)$ be a vector of independently-distributed binomial random variables such that $X\_i\sim \text{Binom}(n,p\_i)$. Further, let $h(\mathbf{X})$ be a convex function on $[0,n]^m$.


**Question:** Is the real-valued function $g(\mathbf{p})=\mathbb{E}\_\mathbf{X}[h(\mathbf{X})\ |\ \mathbf{p}]$ convex on $[0,1]^m$?


*Notes*: 


1. This question is a follow-up from a [previous question](https://mathoverflow.net/questions/122897/is-the-binomial-expectation-of-convex-function-convex-in-p), from which we know that in the univariate case, i.e., $m=1$, $g(p)$ is convex.
2. I have done some numerical testing and it appears that $g(\mathbf{p})$ is convex
3. The function $g(\mathbf{p})$ is identical to [one representation](http://www.iue.tuwien.ac.at/phd/heitzinger/node17.html) of the multivariate Berstein polynomial. I haven't seen this representation much in the literature (see another MO [question](https://mathoverflow.net/questions/124523/two-different-representations-of-multivariate-bernstein-polynomials)). However, there is another more common representation, for which some [convexity results](http://www.sciencedirect.com/science/article/pii/0167839691900316) exist.



Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?
The answer is no, a counterexample is for $n=1$, $m=2$: let $h$ be zero on (0,0), (0,1), (1,0), and let $h(1,1)=1$. It is rather clear that you can get it for a convex $h$.


Then, $g(p\_1,p\_2)=p\_1 p\_2$, which is not a convex function on the unit square.


