# Half-space

For other uses, see Half-space (disambiguation).

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 + ... + anxn > b

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

a1x1 + a2x2 + ... + anxn [itex]\geq[itex] b

specifies a closed half-space.

## Properties

A half-space is a convex set.

Proof:

[itex]S=[itex]{[itex]v:\langle v,u\rangle >c[itex]} is a convex set.

Take x,y in S: => [itex]\langle x,u\rangle >c[itex] and [itex]\langle y,u\rangle >c[itex]

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

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

We have:

[itex]a\langle x,u\rangle > ac[itex]

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

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

=> [itex]a\langle x,u\rangle + b\langle y,u\rangle > c[itex]

Thus, [itex]\langle ax+by,u\rangle > c[itex]

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

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