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.