A continuous map $f$ between two metric spaces is said to be a $r$-map if preimage of each point under $f$ has diameter atmost $r$. Suppose $D^n=\{x\in \mathbb{R}^n\mid ||x||\leq 1\}\subset \mathbb{R}^n$. My question is the following
Question. Is it true that for any $\epsilon>0$ there exists a $\delta>0$ such that any $\delta$-map $f:D^n\rightarrow \mathbb{R}^n$ is homotopic through $\epsilon$-maps to an embedding?
The above question seems to be closely related to Steve Ferry's $\beta$-approximation theorem (https://www.maths.ed.ac.uk/~v1ranick/papers/davenema.pdf (page 427, Theorem 8.2.2)). Suppose $M^n$ and $N^n$ are two $n$-manifolds and $M$ has a metric. In this setting, I think Ferry's theorem implies the following
For any $\epsilon>0$ there exists a $\delta>0$ such that any surjective $\delta$-map $f:(M^n,\partial M)\rightarrow (N^n,\partial N)$ is homotopic through $\epsilon$-maps to a homeomorphism if $n\geq 6$.
In my quesion, I do not have surjectivity assumption of the map and instead of a homotopy to a homeomorphism I am looking for a homotopy to an embedding. If anyone have any insight on this please let me know. Thanks in advance.
