# hyperplane

(redirected from Affine hyperplane)
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## hyperplane

(ˈhaɪpəˌpleɪn)
n
(Mathematics) maths a higher dimensional analogue of a plane in three dimensions. It can be represented by one linear equation
Collins English Dictionary – Complete and Unabridged, 12th Edition 2014 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003, 2006, 2007, 2009, 2011, 2014
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These include the face semigroup of an affine hyperplane arrangement, the set of covectors associated with an affine oriented matroid or a T-convex set of topes and the face semigroup of a finite CAT(0) cube complex.
For each positive root [alpha] [member of] [[PHI].sup.+] and k [member of] Z, we define an affine hyperplane
For an affine hyperplane H = {x [member of] [R.sup.d] | <x, [upsilon]> = c} defined by a normal vector v whose last coordinate is positive, a point a [member of] [R.sup.d] is said to be above (resp.
For an affine hyperplane H and a normal vector direction, let [H.sup.+] be the strictly positive side of H and [H.sup.-] the weakly negative side of H.
An affine hyperplane arrangement splits the ambient space into a set of faces.
An affine hyperplane H in V is a level set of a linear functional on V.
Now let [s.sub.n] denote the reflection in the affine hyperplane [e.sub.1] - [e.sub.n] = 1.
The element [s.sub.0] acts as reflection with respect to the affine hyperplane [H.sub.[theta],1].
If [[sigma].sub.I] and [[sigma].sub.J] are adjacent, the quasi-polynomials [q.sub.I] and [q.sub.J] coincide not only on the affine hyperplane spanned by the facet [bar.[[sigma].sub.I]] [intersection] [bar.[[sigma].sub.J]] but also on close parallel hyperplanes.
Given [alpha] [member of] [PHI] and k [member of] Z, we denote by [s.sub.[alpha],k] the reflection in the affine hyperplane
A (real) hyperplane arrangement is a finite collection of affine hyperplanes in Euclidean space.
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