half-space

In geometry, a half-space is any of the two parts into which a hyperplane divides an affine space. More strictly, an open half-space is any of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A strict linear inequality

a₁x₁ + a₂x₂ + ... + a_nx_n > b

specifies an open half-space, while a non-strict one

a₁x₁ + a₂x₂ + ... + a_nx_n

\geq

specifies a closed half-space.

Properties

A half-space is a convex set. Proof:

S=

{

v:\langle v,u\rangle  >c

} is a convex set. Take x,y in S: =>

\langle x,u\rangle >c

and

\langle y,u\rangle >c

Consider the inner product of (ax+by) and u, where a+b=1.

\langle ax+by,u\rangle = a\langle x,u\rangle + b\langle y,u\rangle

We have:

a\langle x,u\rangle > ac

b\langle y,u\rangle > bc=(1-a)c

a\langle x,u\rangle + b\langle y,u\rangle > ac+(1-a)c = c

a\langle x,u\rangle + b\langle y,u\rangle > c

Thus,

\langle ax+by,u\rangle  > c

This proved that the vector (ax+by) belongs to the set S, hence => S is convex.

Half-space

Properties

See also