Star-like implies contractible

Statement
Any uses property satisfaction of::star-like subset of Euclidean space is a proves property satisfaction of::contractible space. In fact, it has a contracting homotopy to any point in the kernel that has the following additional properties:


 * The homotopy is a defining ingredient::linear homotopy.
 * The homotopy is a defining ingredient::semi-sudden homotopy, i.e., for $$t < 1$$ the map $$x \mapsto F(x,t)$$ is a homeomorphism to its image.
 * The homotopy is a (strong) defining ingredient::deformation retraction.

In particular, a star-like subset of Euclidean space (and more generally, a topologically star-like space) is a proves property satisfaction of::semi-suddenly contractible space as well as a proves property satisfaction of::SDR-contractible space.

Related facts

 * Star-like implies any retraction into the kernel is a strong deformation retraction