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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, 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
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
 divides an affine space
Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin....
.

One can have open and closed half-spaces.






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Half Space
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, 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
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
 divides an affine space
Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine geometry properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin....
.

One can have open and closed half-spaces. An open half-space is either of the two open set
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
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 (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
Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.Linear equations can have one or more variables....
 that specifies the defining hyperplane.

A strict linear inequality
Inequality

In mathematics, an inequality is a statement about the relative size or order of two objects, or about whether they are the same or not *The notation a < b means that a is less than b....


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

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, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
    .
  • Any convex set
    Convex set

    In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
     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 (x1, x2, ..., xn) such that xn > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that xn be negative (non-positive).

See also


  • upper half-plane
    Upper half-plane

    In mathematics, the upper half-plane H is the set of complex numberswith positive imaginary part y.The term is associated with a common visualization of complex numbers with points in the plane endowed with Cartesian coordinates, with the Y-axis pointing upwards: the "upper half-plane" corresponds to the half-plane above the X...
  • Poincaré half-plane model
    Poincaré half-plane model

    In non-Euclidean geometry, the Poincar? half-plane model is the upper half-plane, together with a metric, the Poincar? metric, that makes it a model of two-dimensional hyperbolic geometry....