See Also

Half-space

In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a half-space is any of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is any 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 any of the two open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

s 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 .

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Encyclopedia


In geometry Geometry

Geometry arose as the field of knowledge dealing with spatial relationships.... 

, a half-space is any of the two parts into which a plane divides the three-dimensional space. More generally, a half-space is any 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 any of the two open set Open set

In topology [i] and related fields of mathematics [i], a set [i] U is called open if, intuitively sp ... 

s 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 . A half-space in a one-dimensional Dimension

In common usage, a dimension is a parameter [i] or measurement [i] required to define the characteristi ... 

 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 Inequality

In mathematics [i], an inequality is a statement about the relative size or order of two objects.
... 



a1x1 + a2x2 + ... + anxn > b


specifies an open half-space, while a non-strict one

a1x1 + a2x2 + ... + anxn b


specifies a closed half-space. Here, one assumes that not all of the real numbers a1, a2, ..., an are zero.

Properties


  • A half-space is a convex set Convex set

    In Euclidean space [i], an object is convex if for every pair of points within the object, every point o ... 

    .

Upper and lower half-spaces


The open upper half-space is the half-space of all such that xn >0 . The open lower half-space is defined similarly, by requiring that xn be negative .

See also


  • upper half-plane
  • Poincaré half-plane model






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