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