Affine set - Knowlpedia

Let $X$ be a vector space. A subset $\Omega\subseteq X$ is affine if for all $a,b\in\Omega$ ,

L[a,b]\subseteq \Omega,

where $L[a,b]$ is the line connecting $a$ and $b$ .

Examples

Any linear subspace is affine.
In $\mathbb{R}^n$ , a set of the form $x_0+L$ with $L$ a subspace is affine.
A convex set need not be affine; affine sets are "flat," while convex sets may be curved.

Equivalent characterizations

A nonempty subset $\Omega$ is affine if and only if it is a translate of a linear subspace (see the translate characterization). Under the line-based definition above, the empty set is also affine.