Convex subset of Euclidean space

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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.

Facts

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.