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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 $\geq$ b

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.

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