Boundary-fixing homeomorphism group of disk is transitive on interior points

Statement
Suppose $$D$$ is a closed disk in $$\R^n$$, and let $$\partial D$$ denote the boundary of $$D$$ ($$\partial D$$ is thus a sphere in $$\R^n$$). Then, for any points $$a,b$$ in the interior of $$D$$ (i.e., $$a,b$$ in $$D \setminus \partial D$$), there exists a self-homeomorphism $$\sigma:D \to D$$ such that $$\sigma(a) = b$$ and $$\sigma(x) = x$$ for all $$x \in \partial D$$.

Applications

 * Connected manifold implies homogeneous: Given any two points in a connected manifold, there is a homeomorphism from the manifold to itself that sends the first point to the other.
 * Euclidean implies compactly homogeneous: Given any two points $$a,b \in \R^n$$, there exists a compact subset $$K$$ of $$\R^n$$ and a homeomorphism of $$\R^n$$ sending $$a$$ to $$b$$ that is the identity map outside $$K$$.

The reflection construction
The idea here is to consider, for every $$x \in \partial D$$, the line segments joining $$a$$ to $$x$$ and the line segment joining $$b$$ to $$x$$. The first line segment is reflected onto the second: the reflection occurs in such a way that ratios of lengths are preserved. In particular, $$x$$ gets sent to $$x$$ and $$a$$ gets sent to $$b$$.

The translation construction
This construction is based on the idea that we have a homeomorphism between the interior of $$D$$ and $$\R^n$$ that works radially: it uses a homeomorphism between $$[0,1)$$ and $$[0,\infty)$$ and scales every point along the radial line from the center of the sphere, via this homeomorphism.

The idea is to take the unique homeomorphism of the interior of $$D$$ that corresponds to a translation map in $$\R^n$$ between the points of $$\R^n$$ corresponding to $$a$$ and $$b$$ under this homeomorphism. It turns out that this homeomorphism of the interior of $$D$$ extends to a homeomorphism of $$D$$ that preserves all the boundary points of $$D$$.