Consider $n \geq 2$ and the simplex \begin{equation} \Delta=\{(p_1,\cdots,p_n) \in \mathbb{R}^{n} \mid \forall i, p_i \geq 0 \text{ and } \sum_{i=1}^{n}{p_i}=1\} \end{equation}
Suppose that $\Delta$ is endowed with a (partial) order $\succeq$ such that
\begin{equation*} p \succeq q \Rightarrow p \succeq \alpha p + (1-\alpha) q \succeq q \end{equation*} for all $(p,q) \in \Delta^2, \alpha \in [0,1]$.
Consider four vectors $(p,q,r,t)$ and two scalars $\lambda \in [0,1], \mu \in [0,1]$ such that:
- $p \succeq q \succeq r \succeq t$
- $\lambda p + (1-\lambda) t = \mu q + (1-\mu)r$
I am trying to prove that these conditions are sufficient to guarantee that the random vectors $(\lambda:p;(1-\lambda):t)$ and $(\mu:q;(1-\mu):r)$ are ranked according to the stochastic increasing convex order, i.e. that \begin{equation*} \lambda V(p) + (1-\lambda) V(t) \geq \mu V(q) + (1-\mu) V(r) \end{equation*} for any continuous and convex $v:\Delta \rightarrow \mathbb{R}$ such that $V(p) \geq V(q)$ whenever $p \succeq q$.
The result is obvious when $n=2$ or $q=\alpha p + (1-\alpha) t$ and $r=\beta p + (1-\beta) t$ for some $(\alpha,\beta) \in [0,1]^2$ since these cases boil down to the one-dimensional problem. But I can't find a way to deal with the general case.
Any hint, help or reference would be greatly appreciated. Thank you!
