Given a function f such that:$$f:P \backslash \{p\} \rightarrow S^{1} \space via \space f(q) =\frac{q-p}{|q-p|} $$ given that P is a star-shaped polygon with respect to a point p in the interior of P.
I would like to show that f is a homotopy equivalence.
How should I tackle this statement? The definition of a homotopy equivalence is understandable but how can I apply it to prove that f is a homotopy equivalence.
I am new to the topic so any input will be greatly helpful. Thank you!
Note that the restriction of $f$ to the boundary $\partial P$ of the polygon is a homeomorphism, since it is bijective continuous from a compact to a Hausdorff space. Call $g$ the inverse of this map, sending $S^1$ homeomorphically to $\partial P$.
The composition $gf$ is the projection from the $P\,\backslash\{p\}$ to the boundary of the polygon $\partial P$. First show this is homotopic to the identity - in fact, $\partial P$ is a deformation retract of $P\,\backslash\{p\}$.
The composition $fg$ is the identity by construction.