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I am looking for a reference for the following kind of results. Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm. Let $B$ be a Borel subset of $\Gamma$ for which the following property holds: for every $\gamma \in B$ there exists a function $h_\gamma \colon [0,1] \to [0,T]$ (for some $T>0$ fixed) which is Lipschitz, non-decreasing and such that $$\tag{1} \frac{dh_\gamma}{dt}(t) = 0 \Rightarrow \frac{d \gamma}{dt}(t)=0. $$ Let $s_{\gamma}$ be (any) inverse of $h_\gamma$ ($s_\gamma$ be have jumps and there is arbitrarily defined, taking any value in the interval $(s_\gamma^-,s_\gamma^+)$); in view of the assumption (1) it turns out the the map $$ \tilde{\gamma}(r) := \gamma(s_\gamma(r)) $$ is well defined and continuous. Thus we define a "reparametrization" map $$ R:\Gamma \to C([0,T]; \mathbb R^d) $$ by $R(\gamma)= \tilde \gamma$. If we consider the sup norm also on $C([0,T]; \mathbb R^d)$ we can formulate the

Question. Which is the regularity of $R$? Is it Borel?

I am pretty sure the result is true and well known but I cannot fined any reference nor I am able to prove it in a clean and reasonably quick way. Any ideas? Thanks in advance

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  • $\begingroup$ Not quite clear to me: in these assumptions, $B\ni \gamma\mapsto s_\gamma$ could be any map taking $\gamma$ to some $C^1$ diffeomorphism $s_\gamma$ in any crazy irregular way, possibly having nothing to see with $\gamma$, since (1) would be automatically satisfied. Say that the function $s_\gamma$ is always either $\arctan$ or $\exp$, according whether $\gamma$ is in some ineffable set $C\subset B$ or not. Why should $R$ be any nicer than $\gamma\mapsto s_\gamma$? $\endgroup$
    – Pietro Majer
    Commented Apr 2, 2017 at 14:06
  • $\begingroup$ Thanks for the useful comment. Yes, I have been imprecise. Let me add that I am assuming the association $\gamma \mapsto h_{\gamma}$ is Borel (between the space of Lipschitz curves in $\mathbb R^d$ and the Lipschitz maps in $\mathbb R$). Say now that I define $s_\gamma$ to be $s_\gamma(r) = \inf \{t: h_\gamma(t)>r \}$ (this should be the left inverse of $h_\gamma$). Is the corresponding association $\gamma \mapsto s_\gamma$ now at least Borel? Thanks again. $\endgroup$
    – Romeo
    Commented Apr 2, 2017 at 14:18
  • $\begingroup$ Ok, thank you. Then I would first consider $\gamma\mapsto h_\gamma + {1\over n}\mathrm{id}$, since I guess $s_\gamma$ can be obtained as the limit of $(h_\gamma + {1\over n}\mathrm{id})^{-1}$ for $n\to\infty$... $\endgroup$
    – Pietro Majer
    Commented Apr 2, 2017 at 15:32
  • $\begingroup$ @PietroMajer Thanks again for your kind reply! Exactly, that was also an idea I considered but I got stuck because: 1. I am not completely sure of having an argument to prove that (inverses) of strictly monotone reparameterizations induce Borel maps between curves: do you have any references for this? 2. I had not a clear idea of how the perturbations converge (pointwise?); 3. I thought there has to be a general (well-known) argument behind (thus I came here to ask). Thanks for your valuable comments. $\endgroup$
    – Romeo
    Commented Apr 2, 2017 at 15:40
  • $\begingroup$ Shouldn't the target space of $R$ be maps from $[0,T]$ to $\mathbb{R}^d$? Do you want to restrict the domain of $R$ to $B$? $\endgroup$
    – Rbega
    Commented Apr 3, 2017 at 19:26

1 Answer 1

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Peter Michor's book is exactly what you need, and some of his recent work involves more discussion about reparameterization.

Michor, Peter W. Manifolds of differentiable mappings. 1980.

Some of David Mumford's work also discussed the topic, a user-friedly paper might be

Le Brigant, Alice, Marc Arnaudon, and Frédéric Barbaresco. "Reparameterization invariant metric on the space of curves." International Conference on Networked Geometric Science of Information. Springer International Publishing, 2015.arxiv

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  • $\begingroup$ Thanks a lot for your interest and for your references. I admit I have not a solid background in differential geometry, thus I am a bit lost in the huge book you mention. I went through the third chapter and some of the others but I did not find anything specifically related to my problem. The same for the paper you mention and another paper by Michor I have seen: it seems they do not discuss the regularity (at the level of measurability at least, which is the one I need) of the reparameterization maps. Could you be so kind to indicate me a more precise reference therein, please? Thanks! $\endgroup$
    – Romeo
    Commented Apr 2, 2017 at 14:12
  • $\begingroup$ I am not familiar with regularity in measurable sense, so I am sorry I cannot help in that direction. $\endgroup$
    – Henry.L
    Commented Apr 2, 2017 at 14:42
  • $\begingroup$ Thus I am sorry, but I cannot accept your answer. $\endgroup$
    – Romeo
    Commented Apr 2, 2017 at 15:09
  • $\begingroup$ @user-unknown There is another source of literature that discuss natural parameterization, please see Kurtek, Sebastian, et al. "Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces." Biennial International Conference on Information Processing in Medical Imaging. Springer Berlin Heidelberg, 2011. $\endgroup$
    – Henry.L
    Commented Apr 11, 2017 at 11:59

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