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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).

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

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    $\begingroup$ (sorry I couldn’t resist) $\endgroup$ Aug 10 '21 at 15:01
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    $\begingroup$ 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
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    $\begingroup$ 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

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