Homogeneity lemma in point set topology The statement is simple:

For all $x\in \operatorname{int} D_n:=\{x\in\Bbb R^n: \| x \|<1\}$, there exists a homeomorphism $h$ on $D_n$ such that $h(0)=x$ and $h$ does not move the points of $S_{n-1}$. 

This somehow reminds me of the sound wave pattern present in the Doppler effect: 

Perhaps I can accordingly construct an explicit homeomorphism mapping? But it's a very vague and naive idea, and doesn't seem to help me in any practical way. 
If an explicit homeomorphism is not to be easily found, then is there any intuitive (and rigorous and convincing, of course) way to tackle this seemingly simple problem?
(When I search for a proof, I find that most results are actually not what I want, rather, they seem relevant to another lemma (under the same name) in differential topology/geometry, of which I know nothing.)
Ps: the following theorem might be related (but I don't know how): 

Any closed, convex "volume" (I don't know the terminology; put simply, something solid and has a volume) in $\Bbb R^n$ is homeomorphic to $D_n$. 

 A: This reminds me a result whose proof is very similar to this one :

For a connected manifold $M$ and any two points $x,y$ in $M$ there exists a homeomorphism of $M$ that sends $x$ to $y$.

I'm giving an outline of the proof. Consider the straight line segment joining $ 0$ to $x$ and take a small tubular neighborhood $N$ of this segment in $ D_{n}$ . Consider constant vector field $X$ in $N$ parallel to the line segment. Take another tubular nhbd $M$ inside $N$ containing the line segment.Now get a bump function $ f $ which is $1$ on whole $M$ and $0$ on the complement of $N$.Now $ f.X$ is a vector field on  int$ D_{n}$ . This being a compactly supported vector field is complete. Hence you will get a homeomorphism from int  $ D_{n}$ to itself taking $0$ to $x$. Also this homeomorphism fixes every point outside of $N$ and so it can be extended to whole  $ D_{n}$ fixing the boundary sphere. 
A: There is an elementary approach that works in any normed real vector space.
The general idea is to use linear interpolation along radii.
Concretely, any $x \in D$ can be written as $x = te$ with $t \in [0, 1], \|e\| = 1$. We can set up a general form $h(te) = C + tf(e)$ with unknown
$C$ and $f$, and then try to solve the system $h(0) = a, h(e) = e$. This
works out to the pleasantly simple $h(x) = x + (1 - \|x\|)a$.
Some elementary linear algebra shows that $\|h(x)\| \le 1$ when
$\|x\| \le 1$, and furthermore for arbitrary $x, y$
$$
  (1 - \|a\|)\|x -  y\| \le \|h(x) - h(y)\| \le 2\|x - y \|
$$
so $h$ is at least a homeomorphic embedding.
To show that it is surjective may be a little trickier, since there seems
to be no easy expression for $h^{-1}$, but it can be done by applying the intermediate value theorem to the function $g(t) = \|a + t(x - a)\|$ to
show that there is a unit vector $e$ such that $x$ is in the segment 
$[a, e] = h([0, e])$.
