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.