Let $\Omega$ be a convex body$^{\boldsymbol{1}}$ in $\mathbb{R}^n$ where $n$ is a positive integer. Fix a positive integer $k$ and some $0<\alpha\leq 1$. Let $k_1> k_2>0$. Does there necessarily exist a diffeomorphism $\phi^{k,\alpha}\in C(\Omega,\Omega)$ satisfying: $$ \lVert\phi-1_{\Omega}\rVert_{k,\alpha}= k_1 \text{ and } \lVert\phi-1_{\Omega}\rVert_{\infty}\leq k_2, $$ where $\lVert\cdot\rVert_{k,\alpha}$ is the usual norm on the Hölder space $C^{k,\alpha}(\Omega,\mathbb{R}^n)$ and $\lVert\cdot\rVert_{\infty}$ is the familiar sup-norm on $C(\Omega,\mathbb{R}^n)$.

Intuitively, I imagine this can be constructed by starting with some “small homeomorphism” $\tilde{\phi}:\Omega\rightarrow \Omega$ and then smoothing it out/mollifying it. But I don't know how to formalize this idea, or if it is even true.

Edit$^{\boldsymbol{1}}$: Following @Pietro Majer's point; I should mention that I also assume that $\Omega$ is a convex body in $\mathbb{R}^n$ (so non-empty interior) and that $n\in \mathbb{Z}^+$ (so $\Omega$ cannot be a point).

  • $\begingroup$ There is something missing in the question or in my understanding: if $1_\Omega$ is the identity, then $\phi = 1_\Omega$ would do. $\endgroup$ Aug 10 '21 at 14:34
  • $\begingroup$ @BenoîtKloeckner You're right. I meant to have an equality $\|\phi-1_{\Omega}\|_{k,\alpha}=k_1$ and not an inequality (or else, the problem becomes trivial as you noted). $\endgroup$ Aug 10 '21 at 14:39

As it is the answer is no, by the following counter-example $$.$$

  • 2
    $\begingroup$ (sorry I couldn’t resist) $\endgroup$ Aug 10 '21 at 15:01
  • 3
    $\begingroup$ Is the counterexample a point? $\endgroup$
    – LSpice
    Aug 10 '21 at 15:03
  • $\begingroup$ Oh hahah! It took me a second, ok.. But what is $\Omega$ is say a convex body in $\mathbb{R}^n$ and $n>0$. Is it still obvious $\endgroup$ Aug 10 '21 at 15:45
  • 2
    $\begingroup$ For a convex body, I would take the flow of a sufficiently large(1) smooth field with sufficiently small(2) support in the interior of $\Omega$. The flow $\eta^t$ at time $t$ is a one parameter family of diffeos $\Omega\to\Omega$, that vary continuously wrto $t$, from $\eta^0=Id$ to a distance (in the chosen norm) larger than $k_1$ by (1), yet less than $k_2$ in uniform norm by (2). By continuity some $t$ should do. $\endgroup$ Aug 10 '21 at 16:46
  • $\begingroup$ @PietroMajer That's a great idea, thanks :) $\endgroup$ Aug 10 '21 at 22:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.