Convex subset of Euclidean space

Definition
A convex subset of Euclidean space is a subset in $$\R^n$$ 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 $$\R^n$$, we can also apply this definition to an infinite-dimensional Euclidean space, where it is also very useful.

Any two functions to a convex subset are linearly homotopic
If $$C$$ is a convex subset of $$\R^n$$ 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.