# Convex subset of Euclidean space

## Contents

## Definition

A **convex subset** of Euclidean space is a subset in for some , 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 , 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 is a convex subset of and are continuous functions, then we can define a linear homotopy from to , and hence and are homotopic.

### Any retract is a deformation retract

Suppose is a convex subset and is a retraction (viz, is a continuous map from to such that is the identity map. Then the linear homotopy between the identity map on and the function is a deformation retraction from to .

### 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 , it gives a homeomorphism to its image (since it's just dilation by a factor). Thus, it is a semi-sudden homotopy.