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Let $K$ be a compact subset of $\mathbb{R}^n$ (for simplicity, I am happy to take $K=\overline{B(0,1)}$ for now if it is easier).

Let $f:K \rightarrow \mathbb{R}^m$ be a continuous function.

Is there a new metric $d$ on $\mathbb{R}^m$ (compatible with the topology) such that $$ f: K \rightarrow (\mathbb{R}^m,d) $$ is Lipschitz?

(Here $K$ is still equipped with the usual Euclidean metric, and only the metric on the target space $\mathbb{R}^m$ has changed.)

If not,

  • What are obstructions?
  • Are there sufficient conditions?
user155731
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  • See this question and links therein https://mathoverflow.net/questions/291563/what-is-the-name-for-a-set-endowed-with-a-lipschitz-structure – erz Apr 06 '20 at 15:58
  • Thank you erz. The reference of Luukainen-Vaisala linked there seems relevant, but I have not yet found an answer to this question there (at an admittedly brief glance). If I do, I will update. – user155731 Apr 06 '20 at 16:12
  • also relevant (in fact essentially the same) https://mathoverflow.net/questions/195611/quotient-of-metric-spaces – Pietro Majer Apr 13 '20 at 16:46

3 Answers3

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It can't be done. Here's a counterexample with $n = 2$ and $m = 1$.

For each $k \in \mathbb{N}$, define a function $f_k: [0,1] \to [0,1]$ by linearly interpolating between the values $f_k(\frac{i}{k}) = \frac{i}{k}$ ($0 \leq i \leq k$) and $f_k(\frac{i}{k} + \frac{1}{k^2}) = \frac{i+1}{k}$ ($0 \leq i \leq k -1$). They look like staircases.

I claim there is no metric on the target $[0,1]$ that makes all of the $f_k$ contractions. If $f_k$ is a contraction this forces the distance from $\frac{i}{k}$ to $\frac{i+1}{k}$ to be at most $\frac{1}{k^2}$, and thus the distance from $0$ to $1$ to be at most $\frac{1}{k}$. If this were true for all $k$ then the distance from $0$ to $1$ would have to be zero and the metric would not be compatible with the topology.

Now define $f(t,\frac{1}{k}) = f_k(t)$ for all $k$ and $f(t,0) = t$. Then $f$ is a continuous function defined on a compact subset of $[0,1]^2$, so by Tietze it extends to a continuous function from the unit square to $[0,1]$. If there were a change of metric on the target $[0,1]$ that made $f$ Lipschitz then by scaling there would be a change of metric that made $f$ a contraction, and this would make each $f_k$ a contraction, contradicting the claim.

Nik Weaver
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  • Nice! Do you have any thoughts on sufficient conditions? If not, I'd still happily accept the answer, since this was the main point of my question. – user155731 Apr 06 '20 at 17:29
  • I'm afraid I don't have anything intelligent to say about sufficient conditions ... I will think about it. – Nik Weaver Apr 06 '20 at 17:43
  • I guess for each distinct $s$ and $t$ in the range you want to look at the value ${\rm inf}{|x_1-y_1|+\cdots+|x_r-y_r|: f(x_1) = s$, $f(y_r) = t$, and $f(y_i) = f(x_{i+1})}$. If it's ever zero there's no hope, but if not this value describes a metric on the range which makes $f$ a contraction. It shouldn't be hard to use this to get a metric on $\mathbb{R}^m$ that makes $f$ a contraction. – Nik Weaver Apr 06 '20 at 19:38
  • Yes, this makes sense. Thanks (to both you and Pietro Majer) for the answers. I'm accepting this one just because it was first and you were kind enough to think about the sufficiency, but Pietro Majer's is also worth accepting! – user155731 Apr 06 '20 at 20:11
  • You are welcome! – Nik Weaver Apr 06 '20 at 21:48
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For instance, no equivalent distance on $\mathbb{R}^1$ can make Lipschitz the Cantor function $f:[0,1]\to\mathbb{R}^1$. Suppose by contradiction $d$ is a distance that makes $f$ $L$-Lip.

Let $F_n$ be the closed sets in the usual construction of the cantor set $C=\bigcap_{n\ge\mathbb{N}}F_n$, that is $F_0=[0,1]$ and $F_{n+1}=\frac{1}{3}F_n \cup (\frac{1}{3}F_n+ \frac{2}{3})$.

If $0=x_1<\dots<x_{2^n}=1$ are the endpoints of the components of $F_n$, then $x_{2i}-x_{2i-1}=\Big(\frac{2}{3}\Big)^n$ and $f(x_{2i})=f(x_{2i+1})$ because $f$ is constant on the connected components of $F_n^c$. But then for all $n$ $$d(1,0)=d(f(0),f(1))\le L\sum_{i=1}^{2^n}|x_i-x_{i-1}|=L \Big(\frac{2}{3}\Big)^n$$ so $d(0,1)=0$.

Pietro Majer
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  • Thank you, this is also a very straightforward example. – user155731 Apr 06 '20 at 20:10
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    So the idea of your example is that a Lipschitz function maps measure-zero sets into measure-zero sets, and whatever metric we choose on the target, the Hausdorff measure of an open set $(0,1)$ has to be positive. Perhaps this can be enhanced somehow to get a sufficient condition? – erz Apr 07 '20 at 02:58
  • Exact, that was what I had vaguely in mind, though I could not find the right words :) – Pietro Majer Apr 07 '20 at 08:38
  • @erz Indeed your idea of a sufficient condition in terms of the Hausdorff measure should be the right one. I tried to expand it a bit. – Pietro Majer Apr 09 '20 at 23:08
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Here are some suggestions trying to expand erz's and Nik Weaver's comments. I do not have time to work out the details (although stuck at home arrests like many of us), but I'd be glad if nevertheless they turn out to be useful.

Recall that a set $S\subset\mathbb{R}^m$ is $\mathcal{H}^1$- null iff for all $\epsilon>0$ it has a countable partition $\{S_j\}_{j\in\mathbb{N}}$ such that $\sum_{j\in\mathbb{N}}\rm{diam}(S_j)\le\epsilon$.

Say the that a map $f:A\subset\mathbb{R}^m\to\mathbb{R}^n$ is $\mathcal{H}^1$- Absolutely Continuous iff it is continuous and for any $\mathcal{H}^1$-null set $S\subset A$, the image $f(S)$ is $\mathcal{H}^1$-null in $\mathbb{R}^n$ .

[edit 4.24.20]: as user erz pointed out to me, this definition is too weak (the function in Nik Weaver's counterexample satisfies it). A more suitable definition is maybe: iff it is continuous and for any $\epsilon>0$ there exists $\delta>0$ such that for any $S\subset A$, $\mathcal{H}^1(S)<\delta$ implies that $\mathcal{H}^1 (f(S))<\epsilon$ .

The conjecture should be:

Any $\mathcal{H}^1$-AC map defined on a compact subset $K$ is Lipschitz up to a choice of an equivalent distance on the target space.

We may always assume $n=m$ (via inclusion $K\subset \mathbb{R}^n\subset\mathbb{R}^m$, or $\mathbb{R}^m\subset\mathbb{R}^n$ if needed). Then

  • It does not seem problematic to extend an $\mathcal{H}^1$-absolutely continuous map $f$ defined on a compact $K$ to a $\mathcal{H}^1$-absolutely continuous map $f:\mathbb{R}^n\to\mathbb{R}^n$ that is also a surjective and proper map (it should sufficient to make it locally lipschitz on the complement of $K$, and equal to the identity outside a large ball.

  • For $x,y$ in $\mathbb{R}^n$ define

$$d(x,y):=\inf\{\mathcal{H}^1(S): f(S)\,{ \rm connected},\, \{x,y\}\subset f(S)\}$$

  • Then $d:\mathbb{R}^n\times \mathbb{R}^n\to[0,\infty)$ is clearly symmetric, and satisfies the triangular inequality.

  • To show $d(x,y)=0$ implies $x=y$, is where the hypotheses enter. The idea should be that $d(x,y)=0$ implies the existence of a $\mathcal{H}$-null subset $S$ such that $f(S)$ is connected and $\{x,y\}\subset f(S)$, which forces $x=y$ since $f(S)$ is also $\mathcal{H}$-null. To this end one should start from a minimizing bounded sequence of compact subsets $S_j$ with $\mathcal{H}^1(S_j)\to0$, and $\{x,y\}\subset f(S_j)$. Compact subsets of a given compact are a compact in the Hausdorff distance, the Hausdorff measure is lower semicontinuous, connected sets are a closed set, so I think it could be done. In fact, it seems OK that the infimum in the definition of this distance be always attained, by the same argument.

  • By definition of $d$, taking as $S$ the segment $[u,v]$ one has $d(f(u),f(v))\le \mathcal{H}^1([u,v])= \|u-v\|$.

  • It remains to show that $d$ is topologically equivalent to the Euclidean distance, which seems true, although I see it less clearly at the moment.

Pietro Majer
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  • This goes far beyond my little comment. Very interesting! – Nik Weaver Apr 10 '20 at 02:24
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    Could you please elaborate on the extension? As for topological equivalence, here is a proof if $K$ is "nice". Assume $d(y_n,y)\to 0$ and $\varepsilon>0$. Since $f$ is uniformly continuous on $K$, there is $\delta$ such that $|x-z|<\delta\Rightarrow |f(x)-f(z)|<\varepsilon$. For large $n$ $d(y_n,y)<\delta$, and so there are connected $S_n\subset K$ and $x_n,z_n\in S_n$ such that $f(x_n)=y,~ f(z_n)=y_n$ and $|y_n-z_n|\le H^1(S_n)<\delta$, from where $|y_n-y|\to0$. Assume $|y_n-y|\to0$ but $d(y_n,y)\not\to 0$. WLOG $d(y_n,y)>a>0$. Choose $z_n\in f^{-1}(y_n)$. From compactness WLOG $z_n\to z$. – erz Apr 12 '20 at 03:32
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    If $K$ is nice so that we can join $z_n$ and $z$ with short curves, then we get a contradiction with $d(y_n,y)\not\to 0$. So $K$ should be such that the length metric is equivalent with an Euclidean one. If you indeed can extend $f$ to a closed neighborhood of $K$ that would be enough. – erz Apr 12 '20 at 03:35
  • I'd like to add that this condition is necessary to existence of metric on $f(K)$ such that $f$ is Lipschitz, and also the coordinate projections on $f(K)$ are Lipschitz. Perhaps it is also sufficient. What I am saying is that the condition depends on the Euclidean metric on the image, and is not purely topological, as one would like to. – erz Apr 12 '20 at 04:03
  • @erz very good. For the issue of the extension, one can make it locally Lipschitz, or even smooth, on the complement of $K$: isn't this sufficient? For any $H^1$-null set $S$ in the domain, $f(S\cap K)$ is $H^1$-null by hypothesis, and since $S\setminus K$ is covered by countably many balls on which $f$ is Lipschitz, the set $f(S\setminus K)$ is $H^1$-null as well. – Pietro Majer Apr 12 '20 at 20:04
  • In fact, (once it is OK that $d$ is a distance), on the lines of your argument, since $(K, |\cdot|)$ is compact it is sufficient to show that the identity $(K, |\cdot|)\to (K,d)$ is continuous. That is: if $|u_j-v|\to0$ then $d(u_j,v)\to0$.

    Since $f$ is 1-Lip from $ |\cdot|$ to $d$, we have for all $u,v$ $$d(u,v)\le \inf{ |x-y|: x\in f^{-1}(u), , y\in f^{-1}(y)}.$$ Since $f$ is surjective and proper, if $u_j\to v$we have a $ x_j\in f^{-1}(u_j)\subset K$, $x_j\to y\in f^{-1}(v)$, whence $d(u_j,v)\to0$

     – Pietro Majer Apr 12 '20 at 20:32
  • (Also, it seems to me that we may take everywhere, in the definition of $H^1$-AC map and in the definition of $d(x,y)$, all sets $S$ to be compact. Is this equivalent? It seems that everything works as well; maybe it's more clear). – Pietro Majer Apr 12 '20 at 20:39
  • sorry you lost me on your second comment. And still, i cannot see (most likely due to my ignorance), how to extend $f$ to be locally Lipschitz and surjective on the complement of $K$? (I can see why that would be sufficient). As for compactenss in the definitions - yeah, probably those are the "correct" definitions, because everywhere else in the proof the sets are compact. – erz Apr 12 '20 at 20:47
  • (typo: above I mean: the identity map not on $K$, but on $f(K)$ , or on any closed ball) – Pietro Majer Apr 12 '20 at 20:48
  • To extend $f$ to be locally Lipschitz on the complement: we extend each coordinate of $f$, e.g. by means of a Lipschitz partition of unity in the complement of $K$; this is standard, and we can also make for the extended map $f_i(x)=x_i$ for larve $|x|$. The resulting map is proper, and it is surjective because it has topological degree $1$. (Of course, assuming $n=m$) – Pietro Majer Apr 12 '20 at 21:00
  • To be honest, i cannot reconstruct the partition of unity argument (i'd really appreciate if you point me to something analogous). What i came up with is to approximate $f_i$ with a sequence of polynomials $p_n$, while also enclose $K$ with a sequence $U_n$ of open sets that shrinks to $K$ and $\overline{U_{n+1}}\subset U_n$. Construct $g=p_1$ on $U_1$ \ $U_2$, then extend it to $U_1$ \ $U_3$ so that on $U_2$ \ $U_3$ it approximates $p_2$ and so on. In the end define $g=f_i$ on $K$. The resulting function is continuous and smooth outside of $K$. – erz Apr 13 '20 at 02:03
  • @erz : the partition of unity argument is the usual construction of the Tietze extension thm on metric spaces (as e.g. in Dieudonné, Found. of Mod. Anal.); it gives a loc.lip. function if the partition is such. You can also google "tietze extension theorem in metric spaces" – Pietro Majer Apr 13 '20 at 05:33
  • There are other possibilities of course. The quickest is as follows: take a concave modulus of continuity $\omega$ for $f$ on $K$ (see the wiki article linked below); define the extension as in the link: $f^$ := infimum of all $\omega$-continuous functions that are above $f$ on$K$. The function $f^$ has the same mdc $\omega$ globally, and on each set $\Omega_\epsilon:= {x: d(x,K)\ge\epsilon}$ it is Lipschitz; If I'm not wrong precisely with constant $\omega'_+(\epsilon)$ (right derivative). https://en.wikipedia.org/wiki/Modulus_of_continuity#Subadditive_moduli,_and_extendibility – Pietro Majer Apr 13 '20 at 05:41
  • ... the reason is : because on $\Omega_\epsilon$ all functions in the infimum that defines $f^*$ are $\omega'+(\epsilon)$-Lip, and this because $\omega$ itself is $\omega'+(\epsilon)$-Lip on the interval $[\epsilon,+\infty)$, being concave. – Pietro Majer Apr 13 '20 at 05:48