Suppose I have a set of quasi-concave functions $f_1(x), f_2 (x), \dots, f_N(x)$. I form a convex combination,
$$g(x) = \sum_i c_i f_i(x)$$
where the $c_i \ge 0$, $\sum_i c_i = 1$. Given a fixed set of quasi-concave functions $f_i(x)$, the convex combination $g(x)$ will be quasi-concave only for specific values of the coefficients $c_i$ (for example, if $c_1 = 1$, $c_i = 0, i \ne 1$). In general, are there necessary/sufficient conditions that must be satisfied by the $c_i$ (for a fixed set of functions $f_i$), such that $g(x)$ is quasi-concave?
Example: Here is a non-trivial example. If $N = 2$ and the $f_i$ are Gaussian densities, there are non-trivial conditions. See For example, if $N = 2$ and the $f_i$ are Gaussian densities, some conditions can be given. See https://en.wikipedia.org/wiki/Multimodal_distribution#Mixture_of_two_normal_distributions.
Note: This is different from Convex combination of quasiconvex functions.. I know that the convex combination is not quasi-concave in general. My question is if one can state a set of conditions on the $c_i$ such that the combination is quasi-concave. A trivial condition is that only one of the $c_i$ is non-zero. Is there something more?