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][1], from which we know that in the univariate case ($m=1$) $g(p)$ is convex.
 2. I have done some numerical testing and it appears that $g(\mathbf{p})$ is convex
 3. The binomial expectation is identical to [one representation][2] of the multivariate Berstein polynomial. However, there is another more common representation, for which some [convexity results][3] exist.


  [1]: http://mathoverflow.net/questions/122897/is-the-binomial-expectation-of-convex-function-convex-in-p
  [2]: http://www.iue.tuwien.ac.at/phd/heitzinger/node17.html
  [3]: http://www.sciencedirect.com/science/article/pii/0167839691900316