Consider a continuous and even function $f : \mathbb{R} \rightarrow \mathbb{R}$ vanishing at infinity and strictly decreasing over $[0,\infty)$. In particular, $f$ has a unique global maximum at $0$. A typical example is the Gaussian function.
I am wondering under which (necessary and/or sufficient) conditions on $f$ can we say that any linear combination $g = \sum_{n=1}^N a_n f(\cdot - x_n)$ admits at most $N$ global optima for any $N \geq 1$, $a_n \in \mathbb{R} \backslash\{0\}$ and distinct $x_n$.
Similar conditions for local optima are also welcome if this appears to be simpler.