Let $f: \mathbb{R}^d \to \mathbb{R}^k$ be a function. Under what conditions on $f$ does the following statement hold?
$$ \forall w \in \mathbb{R}^k, \quad x \mapsto w^\top f(x) \text{ is quasiconvex}$$
Here, a function $g: \mathbb{R}^d \to \mathbb{R}$ is quasiconvex if $g(\alpha x + (1 - \alpha)x') \leq \max (g(x),g(x'))$ for all $\alpha \in [0, 1]$ and $x, x' \in \mathbb{R}^d$.
One sufficient condition is for $f$ to be linear, since $x \mapsto w^\top f(x)$ is linear when $f$ is linear. However, I am hoping there is a weaker condition that is still sufficient. A few basic things I have noticed so far: When $k=1$, it is sufficient for $f$ to be quasilinear (i.e., both $f$ and $-f$ are quasiconvex). However, for $k>1$ it is not sufficient for each component $f_i(x)$ to be quasilinear, since the sum of quasilinear functions is not necessarily quasilinear.