Is there any approach to find a good upper-bound on the number of modes (local maximums) in the sum of several quasi-concave functions? What is the tightest upper-bound?
I'm very interested to know whether it is possible to find better bounds by considering some constraints on the functions. For example, in special cases, it can be unimodal. I think it can be shown that the sum of several quasi-concave or even star-unimodal functions that have no common parameters will be star-unimodal. For example, if $f(x)$ and $g(y)$ are star-unimodal on $\mathbb{R}^n$ and $\mathbb{R}^m$, then $f(x)+f(y)$ will be star-unimodal on $\mathbb{R}^{n+m}$. Maybe some constraints, such as $\alpha$-concavity, can be helpful.
Note: Star-unimodality means that the function is non-increasing along rays going away from its global maximum (see Dharmadhikari, S., & Joag-Dev, K. (1988). Unimodality, convexity, and applications. Elsevier.)