Star-like subset of Euclidean space

Definition

A star-like subset of Euclidean space is a nonempty subset of $\mathbb {R} ^{n}$ for some $n$ which contains a point (called a star-point) such that the line segment joining that point to every other point in the subset, lies completely inside the subset. The collection of all star points is termed the kernel, so a star-like subset is essentially a subset with nonempty kernel.

In this light, a convex subset is a subset whose kernel is the whole subset.

Facts

Any map is linearly homotopic to a map into the kernel

Suppose $S$ is a star-like subset with kernel $K$. Then if $f,g:X\to S$ are continuous maps and $g(X)\subseteq K$ then we can define the linear homotopy from $f$ to $g$.

This in turn implies that any two maps to $S$ are related by a homotopy which is a composite of two linear homotopies.

Every retract inside the kernel is a deformation retract

The idea is to construct a linear homotopy between the identity map and the retraction.

The space is contractible in a semi-sudden way

For any star point, we can consider the deformation retraction to that point (the linear homotopy between the identity map and the constant map to that point).