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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?

a06e
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  • First of all—the convex combination requires $\sum_i c_i=1$ as well. But no, there is no set of conditions on $c_i$ alone that can guarantee quasiconcavity, other than the trivial case you cited. Because assume that $c_i>0$ for more than one $i$. Then consider any two quasiconcave functions $f_1,f_2$ such that $f_1(x)+f_2(x)$ is not quasiconcave. Then define $g_1(x)=c_1^{-1}f_1(x)$ and $g_2(x)=c_2^{-1}f_2(x)$; they are quasiconcave as well, but $c_1 g_1(x)+c_2 g_2(x)$ is not. – Michael Grant Apr 05 '18 at 23:24
  • @MichaelGrant Yes, I expect that the condition depends on the $f_i$ as well. Sorry if that wasn't clear in the question. I edited. Also, $\sum_i c_i = 1$ adds nothing, since you can always normalize the $c_i$ without changing the shape of $g(x)$ – a06e Apr 06 '18 at 12:32
  • Yeah but once you place conditions on both the functions and the coefficients it’s basically a meaningless problem. – Michael Grant Apr 06 '18 at 12:33
  • @MichaelGrant I mean, given the functions $f_i(x)$, what conditions must be satisfied by the $c_i$ so that the linear combination is quasiconcave? Is that clear enough? – a06e Apr 06 '18 at 12:35
  • I do think I understand. I just don’t think there is any or actually useful answer. Convexity in all its forms is an extremely fragile property. – Michael Grant Apr 06 '18 at 12:37
  • @MichaelGrant 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. – a06e Apr 06 '18 at 12:39

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