Convex subset of Euclidean space

Definition

A convex subset of Euclidean space is a subset in $\mathbb {R} ^{n}$ for some $n$, with the property that given any two points in the subset, the line segment joining those two points also lies completely within the subset.

Note that in place of a finite-dimensional Euclidean space $\mathbb {R} ^{n}$, we can also apply this definition to an infinite-dimensional Euclidean space, where it is also very useful.

Facts

Any two functions to a convex subset are linearly homotopic

If $C$ is a convex subset of $\mathbb {R} ^{n}$ and $f,g:X\to C$ are continuous functions, then we can define a linear homotopy from $f$ to $g$, and hence $f$ and $g$ are homotopic.

Any retract is a deformation retract

Suppose $C$ is a convex subset and $r:C\to D$ is a retraction (viz, $r$ is a continuous map from $C$ to $D$ such that $r|_{D}$ is the identity map. Then the linear homotopy between the identity map on $C$ and the function $r$ is a deformation retraction from $C$ to $D$.

The space is contractible in a semi-sudden way

A convex subset can be contracted to any point in it, by the linear homotopy. The linear homotopy has the further interesting property that for $t<1$, it gives a homeomorphism to its image (since it's just dilation by a factor). Thus, it is a semi-sudden homotopy.