For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal.
From Wikipedia
$f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ such that $f(G(Z))$ is convex.
Also from Wikipedia:
the super-level sets $L(f, t)$ of $f$, defined by $ L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \}, $ are convex subsets of $\mathbb{R}^n$ for every $t ≥ 0$. (This property is sometimes referred to as being unimodal.)
Added: By the comment below, a common definition is that a function is unimodal, if it has exactly one local maximum.
I wonder if this common definition allows existence of more than one local minimum? For example a "W" shape function defined on $\mathbb{R}$ goes to $\infty$ when approaching $\pm \infty$ in domain.
If this definition does not allow existence of any local minimum, then for any line through the mode $m \in \mathbb{R}^n$ of $f$ in its domain, does the restriction of $f$ to this line increase on one side of $m$ and decrease on the other side of $m$?
I wonder if the first two definitions agree with each other? If not, when?
What is your definition, if possible?
Thanks and regards!