HALF-SPACE

In geometry, a 'half-space' is either of the two parts into which a plane divides the three-dimensional space. More generally, a 'half-space' is either of the two parts into which a hyperplane divides an affine space.
One can have ''open'' and ''closed'' half-spaces. An 'open half-space' is either 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 in a one-dimensional space is called a 'ray'.
A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane.
A strict linear inequality
:''a''₁''x''₁ + ''a''₂''x''₂ + ... + ''a''_''n''''x''_''n'' > ''b''
specifies an open half-space, while a non-strict one
:''a''₁''x''₁ + ''a''₂''x''₂ + ... + ''a''_''n''''x''_''n'' $geq$ ''b''
specifies a closed half-space. Here, one assumes that not all of the real numbers ''a''₁, ''a''₂, ..., ''a''_''n'' are zero.

Contents

Properties

Upper and lower half-spaces

See also

Properties

★ A half-space is a convex set.

★ Any convex set can be described as the (possibly infinite) intersection of halfspaces

Upper and lower half-spaces

The open (closed) 'upper half-space' is the half-space of all (''x''₁, ''x''₂, ..., ''x''_n) such that ''x''_''n'' >0 ( ≥ 0). The open (closed) 'lower half-space' is defined similarly, by requiring that ''x''_''n'' be negative (non-positive).

See also

★ upper half-plane

★ Poincaré half-plane model