Half-space
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- 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