Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a homogeneous function of order one for $i=1,\dots,n$. Further suppose that for any $y\in \mathbb{R}^n$ where the first entry is zero (i.e., $y(1)=0$), $f_i(y)=0$ for all $i$.

Consider the following optimization problem: \begin{equation} \begin{aligned} \max_{\{x_i\in \mathbb{R}^n\}} \quad & \sum_i f_i(x_i) \\ s.t. \quad & \sum_i x_i =b \\ & x_i \geq 0 \end{aligned} \end{equation}

Is it the case that at the optimal solution we generically have $x_i=0$ for all but one $i$? Observe that this trivially holds if $f_i$ are linear.

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