# Definition of a function being unimodal

For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal.

1. From Wikipedia

$f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ such that $f(G(Z))$ is convex.

2. 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.)

3. 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!

• I suspect that Definition 1 has a typo: "$f(G(Z))$ is concave" would be more appropriate in the context of unimodality. Either way, the definitions are rather different. Personally, I prefer to use "unimodal" in the sense "has exactly one point of local maximum". But you are free to use it in any sense you wish as long as you precisely state your definition. And if your definition of "unimodal" is uncommon, you should pay it extra.
– user31373
Commented May 25, 2012 at 2:27
• I would concur with @LeonidKovalev; albeit the sense I have seen it used (line search) was having exactly one local minimum. Commented May 25, 2012 at 3:39
• @copper.hat: Thanks! (1) I wonder if you mean having exactly one local maximum, instead of having exactly one local minimum, for definition of unimodality? (2) If it is maximum in (1), does having exactly one local maximum allows having more than one local minimum?
– Tim
Commented May 25, 2012 at 13:38
• My only exposure to the term was with line search algorithms for optimization. It was a convenient assumption (one local min.) to prove some convergence properties. Commented May 25, 2012 at 15:10
• @copper.hat: Does that definition you were exposed to allow existence of local maximum(s)?
– Tim
Commented May 25, 2012 at 16:00