Homogeneity lemma in point set topology

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.

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 $[a, e] = h([0, e])$.