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Call trajectory any continuous function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n$ (here, $\mathbb{R}_{\geq 0}$ is interpreted as time).

A polyhedral partition of $\mathbb{R}^n$ is a finite set of disjoint polyhedra whose union is $\mathbb{R}^n$.

Say that a trajectory is well-behaved if, for all polyhedral partitions of $\mathbb{R}^n$ and all bounded intervals $[a,b] \subseteq \mathbb{R}_{\geq 0}$, $f$ changes polyhedron only a finite number of times during $[a, b]$.

Is this well-behavedness notion equivalent to some known calculus notion?

In other words, do you know how to remove the "for all partitions" from the definition of "well-behaved"?

Even a stronger but general property of functions would help. At the moment I know that well-behavedness is not implied by smoothness and not implied by Lipschitz continuity.

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  • $\begingroup$ I don't see how $\mathbb R^n$ can be covered by finitely many disjoint polyhedra. What is your definition of polyhedra? $\endgroup$
    – Squala
    Commented Dec 16, 2021 at 13:18
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    $\begingroup$ I thought a polytope is a bounded ployhedron (and a polyhedron is an intersection of a finite number of half-spaces). $\endgroup$
    – Dirk
    Commented Dec 16, 2021 at 13:31
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    $\begingroup$ Since polyhedra are defined as intersections of half spaces, wouldn't it be enough to demand that there is no hyperplane that is crossed infinitely often? $\endgroup$
    – Dirk
    Commented Dec 16, 2021 at 13:33
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    $\begingroup$ For stronger properties, maybe worth considering polynomial or real-analytic trajectories? $\endgroup$
    – Martin M. W.
    Commented Dec 16, 2021 at 18:08
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    $\begingroup$ A simple counter-example, which is maybe useful to keep in mind while searching for a general characterization, is the (polar form) spiral $r=e^{\phi}$ with $\phi=-\frac{1}{t}$. Then the trajectory is not well-behaved on $[0,t]$ for every polyhedral partition of $\mathbb{R}^2$ having a vertex at the origin. I mean that the curve passes at the origin for $t=0$, of course. $\endgroup$
    – Alessandro Della Corte
    Commented Jan 3, 2022 at 23:01

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If:

  1. by polyhedron you mean the (possibly unbounded but non-degenerate) intersection of finitely many half-spaces (notice that this implies convexity);

  2. by "changes polyhedron" you mean passes from the interior of one polyhedron to the interior of another polyhedron;

then notice that an analytic curve (meaning a map $f:\mathbb{R}_0^+\to \mathbb{R}^n$ such that every component is a real-analytic function) intersects transversally every compact subset of every hyperplane finitely many times, and therefore has the property you are requiring. Of course asking for analyticity is too much, as there are $C^\infty$, non-analytic curves which also satisfy the request - think about the graph of a (unimodal) bump function.

My impression is that there is no nice complete characterization in terms of already known concepts.

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