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The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors. The product of the transformations in the two hyperplanes is a rotation whose axis is the subspace of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
The (closed) set of points P between two distinct, parallel hyperplanes in R n is called a plank, and the distance between the two hyperplanes is called the width of the plank, w(P). Tarski conjectured that if a convex body C of minimal width w ( C ) was covered by a collection of planks, then the sum of the widths of those planks must be at ...
A convex set (in pink), a supporting hyperplane of (the dashed line), and the supporting half-space delimited by the hyperplane which contains (in light blue).. In geometry, a supporting hyperplane of a set in Euclidean space is a hyperplane that has both of the following two properties: [1]
The set of all points on a line, called a projective range, has as its dual a pencil of lines, the set of all lines on a point, in two dimensions, or a pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes, a double wedge. [5]
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
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.
In mathematics, a hyperplane section of a subset X of projective space P n is the intersection of X with some hyperplane H.In other words, we look at the subset X H of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace.
When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with "compatible" topologies: for example, when for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ ...