A half-space in $mathbf{R}^2$

Consider the set in $mathbf{R}^2$ defined by a single affine inequality:

$mathbf{H} = left{ x ~:~ 2x_1 + x_2 le 3 right},$

This set is a half-space, that is, a set of the form

$mathbf{H} = left{ x ~:~ a^Tx le b right},$

with

a = left( begin{array}{c} 2 1 end{array}right), ;; b = 3.

The half-space ${x ::: 2x_1 + x_2 le 0}$ . The vector a=(2,1) points outwards the set, at a 90^o angle. The set is comprised of points forming an obtuse angle with . The boundary is the hyperplane defined by the single linear equation 2x_1+x_2 = 0 . This is the subspace of vectors orthogonal to .

The half-space ${x ::: 2x_1 + x_2 le 3}$ . The vector a=(2,1) points outwards the set, at a 90^o angle. The boundary is the hyperplane defined by the single linear equation 2x_1+x_2 = 3 .