Linear homotopy

Definition
Suppose $$U$$ is a subset of a (possibly infinite-dimensional) Euclidean space and $$f,g:X \to U$$ are continuous maps. Suppose further that for every $$x \in X$$, the line segment joining $$f(x)$$ to $$g(x)$$ lies completely inside $$U$$. The linear homotopy between $$f$$ and $$g$$ is a homotopy defined as follows:

$$x \mapsto (1-t) f(x) +tg(x)$$

where the computation on the right side is in $$\R^n$$. Essentially we are moving from $$f(x)$$ to $$g(x)$$ along a straight line.

A composite of several linear homotopies is termed a piecewise linear homotopy. If there exists a piecewise linear homotopy between two functions $$f,g:X \to U$$ then we say that $$f$$ and $$g$$ are piecewise linearly homotopic maps.

Facts
One nice thing about linear homotopies is that they do not unnecessarily move points. In other words, if $$f(x) = g(x)$$ for some point $$x$$, the linear homotopy from $$f$$ to $$g$$ fixes $$x$$ at every point. Linear homotopies are thus useful for showing that given retracts are deformation retracts.

Linear homotopies are commonly seen in the following kinds of sets:


 * Convex subset of Euclidean space: Any two functions to such a set are linearly homotopic
 * Star-like subset of Euclidean space: Any two functions to such a set are homotopic via a composite of at most two linear homotopies
 * Compact retract of open subset of Euclidean space