In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by ThøgerBang ( 1950 , 1951 ). [1]
Given a convex body C in Rn and a hyperplane H, the width of C parallel to H, w(C,H), is the distance between the two supporting hyperplanes of C that are parallel to H. The smallest such distance (i.e. the infimum over all possible hyperplanes) is called the minimal width of C, w(C).
The (closed) set of points P between two distinct, parallel hyperplanes in Rn 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 least w(C). That is, if P1,…,Pm are planks such that
then
Bang proved this is indeed the case.
The name of the problem, specifically for the sets of points between parallel hyperplanes, comes from the visualisation of the problem in R2. Here, hyperplanes are just straight lines and so planks become the space between two parallel lines. Thus the planks can be thought of as (infinitely long) planks of wood, and the question becomes how many planks does one need to completely cover a convex tabletop of minimal width w? Bang's theorem shows that, for example, a circular table of diameter d feet can't be covered by fewer than d planks of wood of width one foot each.
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
In machine learning, support vector machines are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues SVMs are one of the most robust prediction methods, being based on statistical learning frameworks or VC theory proposed by Vapnik and Chervonenkis (1974). Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples to one category or the other, making it a non-probabilistic binary linear classifier. SVM maps training examples to points in space so as to maximise the width of the gap between the two categories. New examples are then mapped into that same space and predicted to belong to a category based on which side of the gap they fall.
In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set
In geometry, a curve of constant width is a simple closed curve in the plane whose width is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve.
In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensional space, forms a Kakeya set. Much of the research in this area has studied the problem of how small such sets can be. Besicovitch showed that there are Besicovitch sets of measure zero.
In mathematics, especially in functional analysis, the Tsirelson space is the first example of a Banach space in which neither an ℓ p space nor a c0 space can be embedded. The Tsirelson space is reflexive.
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.. These intersection subspaces of A are also called the flats ofA. The intersection semilattice L(A) is partially ordered by reverse inclusion.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
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 another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.
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In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rn−1 is a minimal surface in Rn, does this imply that the function is linear? This is true in dimensions n at most 8, but false in dimensions n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.
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Combinatorial Geometry in the Plane is a book in discrete geometry. It was translated from a German-language book, Kombinatorische Geometrie in der Ebene, which its authors Hugo Hadwiger and Hans Debrunner published through the University of Geneva in 1960, expanding a 1955 survey paper that Hadwiger had published in L'Enseignement mathématique. Victor Klee translated it into English, and added a chapter of new material. It was published in 1964 by Holt, Rinehart and Winston, and republished in 1966 by Dover Publications. A Russian-language edition, Комбинаторная геометрия плоскости, translated by I. M. Jaglom and including a summary of the new material by Klee, was published by Nauka in 1965. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
In differential geometry, a hedgehog or plane hedgehog is a type of plane curve, the envelope of a family of lines determined by a support function. More intuitively, sufficiently well-behaved hedgehogs are plane curves with one tangent line in each oriented direction. A projective hedgehog is a restricted type of hedgehog, defined from an anti-symmetric support function, and forms a curve with one tangent line in each direction, regardless of orientation.