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A half-space can be either open or closed. 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. The open (closed) upper half-space is the half-space of all (x 1, x 2, ..., x n) such that x n > 0
The lower half-plane is the set of points (,) with < instead. Arbitrary oriented half-planes can be obtained via a planar rotation . Half-planes are an example of two-dimensional half-space .
Half-space (geometry), either of the two parts into which a plane divides Euclidean space (Poincaré) Half-space model, a model of hyperbolic geometry using a Euclidean half-space; Siegel upper half-space, a set of complex matrices with positive definite imaginary part; Half-space (punctuation), a spacing character half the width of a regular space
The Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line B and the half plane is the upper half (y > 0 ) of this plane. Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.
Conversely, if is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then is a convex set, and is the intersection of all its supporting closed half-spaces. [2] The hyperplane in the theorem may not be unique, as noticed in the second picture on the right.
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