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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 halfspace 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.
See also
upper half-plane, Poincaré half-plane model
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