In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S.
Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice.
The intersection semilattice of A, written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These intersection subspaces of A are also called the flats ofA. The intersection semilattice L(A) is partially ordered by reverse inclusion.
If the whole space S is 2-dimensional, the hyperplanes are lines; such an arrangement is often called an arrangement of lines. Historically, real arrangements of lines were the first arrangements investigated. If S is 3-dimensional one has an arrangement of planes.
A hyperplane arrangement in space
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In geometry and combinatorics, an arrangementofhyperplanes is an arrangementof a finite set A ofhyperplanes in a linear, affine, or projective space...
hyperplanes are the (n − 1)-dimensional flats, each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion...
form a monoid under composition. The product of faces of an arrangementofhyperplanes. A left identity of a semigroup S (or more generally, magma) is...
if and only if the vector b lies in the image of the linear transformation A. Arrangementofhyperplanes Iterative refinement Coates graph LAPACK (the...
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. A real hyperplanearrangement A = { H 1 , … , H n } {\displaystyle {\mathcal {A}}=\{H_{1},\ldots ,H_{n}\}} is a finite set ofhyperplanes in R d {\displaystyle...
higher dimensions, i.e. for certain arrangementsofhyperplanes, the alternating sum of volumes cut out by the hyperplanes is zero. Compare with the ham sandwich...
supersolvable arrangement is a hyperplanearrangement that has a maximal flag consisting of modular elements. Equivalently, the intersection semilattice of the...
interpretation of Whitney numbers through arrangementsofhyperplanes, zonotopes, non-Radon partitions, and orientations of graphs". Transactions of the American...
lattice of flats of a graphic matroid can also be realized as the lattice of a hyperplanearrangement, in fact as a subset of the braid arrangement, whose...
set of ν ( d ) {\displaystyle \nu (d)} hyperplanes in general position in d-dimensional real projective space form an arrangementofhyperplanes in which...
Orlik and Louis Solomon, a pioneer of the theory ofarrangementsofhyperplanes. He was awarded a Mathematical Society of Japan Algebra Prize in 2010. Terao...
problem; their algorithm generates all of the vertices of a convex polytope or, dually, of an arrangementofhyperplanes.[AF92][AF96] Birth year from VIAF...
One important example is the case of arrangements of hyperplanes. An arrangementof n hyperplanes defines O(nd) cells, but point location can be performed...
{\displaystyle d} , considering arrangementsofhyperplanes, the complexity of the zone of a hyperplane h {\displaystyle h} is the number of facets ( d − 1 {\displaystyle...
shortness exponent of polytopes. Chapter 18 studies arrangementsofhyperplanes and their dual relation to the combinatorial structure of zonotopes. A concluding...
complements to hypersurfaces in projective spaces and the topology ofarrangementsofhyperplanes. In the early 90s he started work on interactions between algebraic...
interpretation of Whitney numbers through arrangementsofhyperplanes, zonotopes, non-Radon partitions, and orientations of graphs", Transactions of the American...
complement of the complexification of the arrangementof its reflecting hyperplanes; the generalized braid group of W is the fundamental group of the quotient...
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