Piecewise linear homotopy theorems

Statement

This is a list of basic theorems on the existence of piecewise linear homotopies between functions to subsets of $\mathbb {R} ^{n}$:

• Any two functions to a convex subset of Euclidean space are linearly homotopic. The obivous linear homotopy between them works, because the line segment joining any two points in the convex subset is also in the convex subset.
• Any function to a star-like subset of Euclidean space of $\mathbb {R} ^{n}$ is linearly homotopic to the constant function mapping everything to a star point of the subset. The obvious linear homotopy works again, because the line segment joining a star point to any point is completely inside the subset.

Thus, in particular, we can go from any function to any other by composing two linear homotopies (via the star point).

• Any compact subset of Euclidean space that is a retract of an open set containing it, has the property that there exists an $\epsilon >0$ such that any two functions $f$ and $g$ such that $d(f(x),g(x))<\epsilon \ \forall \ x$, are linearly homotopic. Conversely, given a homotopy between two functions to such a compact subset, we can also construct a piecewise linear homotopy with each of the pieces moving functions to new function that are only $\epsilon$-far away.