Affine Set

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

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

Equivalently, $\Omega$ is affine if it is a translate of a linear subspace (see the translate characterization ).

Examples:

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