If $f : \mathbb{R}^n \to \Bbb R$ is monotone over all lines, can it be written as $h \circ l$ where $h$ is monotone and $l$ is a linear form?

Question

Let be $f : \Bbb R^n \to \Bbb R$ monotone over all lines (not affine ones, but if there is an answer over affine lines, I'm interested.)

Is it possible to find $h : \mathbb{R} \to \Bbb R$ monotone and $l : \Bbb R^n \to \Bbb R$ linear so that $f = h \circ l$ ?

I tried to look by supposing I have such a factorization, and as $\ker l$ is a hyperplane, I have $n - 1$ lines where $f$ is constant. I tried to use the monotonicity condition by trying to compare $f(0)$ and over lines, but it didn't work.

The condition you've written doesn't make much sense, since it's equivalent to $$\forall (u',v)\in\Bbb R^n\times \Bbb R^n,\forall (t,t')\in\Bbb R^2,(t\le t'\to f(u't+v)\ge f(u't'+v))$$ by evaluating yours with $u:=-u'$. — , Jun 28 '19 at 14:31
You can simply say that a map from a line to line is monotone if it preserves betweenness. — Moishe Kohan, Jun 28 '19 at 14:42
@MoisheKohan So basically if you have two points, all points between must have their images between the images of the points? — Raito, Jun 28 '19 at 17:22
I suggest you clarify what a ‘line’ means in your question. Do you mean an affine line or a line through the origin as in the accepted answer. For affine lines your question is still open. — Moishe Kohan, Jun 30 '19 at 11:47

Martín-Blas Pérez Pinilla · Accepted Answer · 2019-06-30T09:45:35.713

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Your factorization implies that $f$ is measurable (see $f: \mathbf{R} \rightarrow \mathbf{R}$ monotone increasing $\Rightarrow$ $f$ is measurable). But you can construct a non measurable $f$ monotone over all lines.

EDIT 1: Construction of $f$.

Take $$\varphi:{\Bbb R}^{n - 1}\longrightarrow{\Bbb R}$$ non measurable, and positive, $\eta$ monotone and positive. The function $$g(x) = \varphi(x_1,\dots,x_{n-1}) + \eta(x_n)$$ will be non measurable. Define $f$ in the upper half-space via a stereographic-projection-like bijection transforming the half-lines with constant $(x_1,\dots,x_{n-1})$ (domain of $g$) in rays (domain of $f$). Define $f(0) = 0$. Define $f$ using symmetry in the open lower half-space. Complete $f$ in the rest of space respecting your condition. Done.

EDIT 2: The bijection.

$$\sigma:(x_1,\dots.x_n)\longmapsto\frac{(x_1,\dots,x_{n-1},1)}{\|(x_1,\dots,x_{n-1},1)\|}\,x_n,$$ $f$ in the open upper half-space: $$f = g\circ\sigma^{-1}.$$

edited Jun 30 '19 at 09:45

answered Jun 28 '19 at 20:58

Martín-Blas Pérez Pinilla

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How to construct such a $f$ ? – Raito Jun 28 '19 at 21:03
@Raito, see the edit. – Martín-Blas Pérez Pinilla Jun 29 '19 at 11:39
If you have time to do the "projection-like bijection" or an explicit formula (you don't have to do both), that'd be great, but I understood most of it, thank you! – Raito Jun 29 '19 at 15:28
Could you provide more detail: I did not understand which function you are using and why it is monotonic on every line. Incidentally, the question is unclear on what a "line" means. My reading is that it means an affine line (rather than a 1-dimensional linear subspace). – Moishe Kohan Jun 30 '19 at 01:04
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@MoisheKohan, see the edit for the explicit formula. The supposed consequence $f = h\circ l$ requires that "line" be interpreted as linear 1D-subspace. – Martín-Blas Pérez Pinilla Jun 30 '19 at 09:49
I see. The real question I think, is about affine lines. – Moishe Kohan Jun 30 '19 at 11:45

If $f : \mathbb{R}^n \to \Bbb R$ is monotone over all lines, can it be written as $h \circ l$ where $h$ is monotone and $l$ is a linear form?

1 Answers1