Let $\Delta_n=\{x \in \mathbb R^{n+1}|x_0+x_1+\dots+x_n=1,\,x_i\geq 0\}$ be the standard $n$-simplex with expected value defined $E[X]=\sum k x_k$. Also let $P \in \Delta_n$ and $B(\theta)$ a linear map: $B(\theta) : P \rightarrow Q$ such that $Q \in \Delta_n$ and $E[Q]= \theta E[P] $
What are the conditions on $B(\theta)$ to guarantee that for any $Q^* \in \Delta_n$ and valid $0 \le \theta \le 1$, $E[Q^*] \le \theta n$, there exists a $P^*$ such that $Q^*=B(\theta) P^*$ ?