I'm working on an engineering problem and I manage to reduce it to the following claim, but I'm not sure if it is true. It will be great if someone can give me some ideas!

Let $u(x)$ is an increasing and concave function such that $u(0)=0$.

Let $X_1,..X_n$ be n random variables independently and identically distributed with support $S\subseteq[0,\infty)$, and define $Y_j=r_j X_j$ where $0<r_1\leq r_2 \leq...\leq r_n$ are given.

Problem: Is it always possible to find a probability vector $p\in\mathbb{R}^n$ (i.e $p\geq0$ and $\sum_{i=1}^{n} p_i = 1$), such that $\forall i$: $\mathbb{E}[u(p_i\sum_{j=1}^{n} Y_j)-u(Y_i)]>0$

Idea: Maybe this helps: I can prove that the claim is true IF the next claim is also true (but clearly I don't know if it is): $\forall p \exists i$: $\mathbb{E}[u(p_i\sum_{j=1}^{n} Y_j)-u(Y_i)]>0$

Notes: I believe it is possible, and I have tried using Jensen, Karamata, subadditivity, and Sperner's lemma, but none of these work. Any idea will be super welcome. Thanks!


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