Hyperplanes
A hyperplane is a set described by a single scalar product equality. Precisely, an hyperplane in is a set of the form
where , , and are given.
When , the hyperplane is simply the set of points that are orthogonal to ; when , the hyperplane is a translation, along direction , of that set.
If , then for any other element , we have
Hence, the hyperplane can be characterized as the set of vectors such that is orthogonal to :
Hyperplanes are affine sets, of dimension (see the proof here). Thus, they generalize the usual notion of a plane in . Hyperplanes are very useful because they allows to separate the whole space in two regions. The notion of half-space formalizes this.
Example:
Projection on a hyperplane
Consider the hyperplane , and assume without loss of generality that is normalized ( ). We can represent as the set of points such that is orthogonal to , where is any vector in , that is, such that . One such vector is .
By construction, is the projection of on . That is, it is the point on closest to the origin, as it solves the projection problem
Indeed, for any , using the Cauchy-Schwartz inequality:
and the minimum length is attained with .
Geometry of hyperplanes
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Geometrically, an hyperplane , with , is a translation of the set of vectors orthogonal to . The direction of the translation is determined by , and the amount by .
Precisely, is the length of the closest point on from the origin, and the sign of determines if is away from the origin along the direction or . As we increase the magnitude of , the hyperplane is shifting further away along , depending on the sign of . In the image on the left, the scalar is positive, as and point to the same direction.
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Half-spaces
A half-space is a subset of defined by a single inequality involving a scalar product. Precisely, an half-space in is a set of the form
where , , and are given.
Geometrically, the half-space above is the set of points such that , that is, the angle between and is acute (in ). Here is the point closest to the origin on the hyperplane defined by the equality . (When is normalized, as in the picture, .)
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