# Name for a "location" property of a function

Let $$L$$ be an operator that maps any well-behaved (with bounded integral) function $$f:\mathbb{R} \rightarrow\mathbb{R}$$ to a number $$L_f\in\mathbb{R}$$, such that if $$g(x)=f(x-d)$$ for all $$x\in\mathbb{R}$$, then $$L_g=L_f+d$$.

Subject to the above condition, $$L_f$$ could be, for example:

• the centroid $$\frac{\int_{-\infty}^\infty xf(x)dx}{\int_{-\infty}^\infty f(x)dx}$$
• $$\inf \arg \max_x⁡{f(x)}$$
• the median
• $$\inf \arg \max_x⁡{f'(x)}$$

or any other domain point serving as some kind of "anchor" to the function's "signature", such that it "moves along" when the function is translated.

Is there a name for such a property/operator in general?

• I like this idea but the $\arg\max_x$ examples don't seem to work -- what do you do when the max is not unique? Maybe you could define instead $\inf \arg\max_x f(x)$.
– 6005
Sep 24, 2019 at 11:01
• 6005 yes indeed Sep 24, 2019 at 11:08
• Centroid may fail to exist, such as for the pdf of the Cauchy distribution. Sep 26, 2019 at 13:12

I don't suspect there is a standard term, but I might call such a map $$L$$ horizontal-translation-preserving because it preserves horizontal translation of the input function by mapping it to translation of the output real number. Formally, $$L: (\mathbb{R} \to \mathbb{R}) \to \mathbb{R}$$ preserves horizontal translation (also called "shifting") if it satisfies $$L(x \mapsto f(x - d)) = L(f) + d.$$
(I would prefer just "translation-preserving", but one has to distinguish between translation in the $$x$$- and $$y$$-direction of a function.)
Note that such a map $$L$$ need not be a linear operator (see e.g. the $$\inf \arg \max$$ example), and it also need not be a bounded linear operator, which are the objects normally studied in functional analysis. So I would be careful when saying "let $$L$$ be an operator" that you clarify you don't mean linear or bounded linear.