Consider the following convex program:
\begin{align*} \min g(x) && \text{such that} \\ f_i(x) &\geq b_1 && \text{ for } i \in 1,\ldots,n; \\ f_i(x)+f_j(x) &\geq b_1+b_2 && \text{ for } i,j \in 1,\ldots,n; \\ f_i(x)+f_j(x)+f_k(x) &\geq b_1+b_2+b_3 && \text{ for } i,j,k \in 1,\ldots,n; \\ \cdots \\ f_1(x)+\cdots+f_n(x) &\geq b_1+\cdots +b_n \end{align*}
That is: every function $f_i$ is at least $b_1$; every pair of functions sum up to at least $b_1+b_2$; every three functions sum up to at least $b_1+b_2+b_3$; etc. The $b_i$ are constants: $0<b_1<\cdots <b_n$.
The problem is that the number of constraints is exponential in $n$. Is there a way to attain the same outcome with a convex program of size polynomial in $n$?