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](https://en.wikipedia.org/wiki/H%C3%B6lder_condition#H%C3%B6lder_spaces) $C^{k,\alpha}(\Omega,\mathbb{R}^n)$ and $\lVert\cdot\rVert\_{\infty}$ is the familiar [sup-norm](https://en.wikipedia.org/wiki/Uniform_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).*



Existence of a Hölder homeomorphism satisfying prescribed norm constraints
As it is the answer is no, by the following counter-example
$$.$$


