Half-space

From Academic Kids

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 <math>\geq<math> b

specifies a closed half-space.

Properties

A half-space is a convex set.

Proof:

<math>S=<math>{<math>v:\langle v,u\rangle >c<math>} is a convex set.

Take x,y in S: => <math>\langle x,u\rangle >c<math> and <math>\langle y,u\rangle >c<math>

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

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

We have:

<math>a\langle x,u\rangle > ac<math>

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

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

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

Thus, <math>\langle ax+by,u\rangle > c<math>

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

See also

upper half-plane, Poincaré half-plane model Template:Maths-stub

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