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Hadwiger's Theorem
In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in \R^n. It was proved by Hugo Hadwiger. Introduction Valuations Let \mathbb^n be the collection of all compact convex sets in \R^n. A valuation is a function v : \mathbb^n \to \R such that v(\varnothing) = 0 and for every S, T \in \mathbb^n that satisfy S \cup T \in \mathbb^n, v(S) + v(T) = v(S \cap T) + v(S \cup T)~. A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if v(\varphi(S)) = v(S) whenever S \in \mathbb^n and \varphi is either a translation or a rotation of \R^n. Quermassintegrals The quermassintegrals W_j : \mathbb^n \to \R are defined via Steiner's formula \mathrm_n(K + t B) = \sum_^n \binom W_j(K) t^j~, where B is the Euclidean ball. For example, W_o is the volume, W_1 is proportional to the surface measure, W_ is proportional to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (geometry)
In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set X with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space \R^n. Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra. Definition Let X be a set and \mathcal S be a collection of admissible subsets of X. A function \phi on \mathcal S with values in an abelian semigroup R is called a valuation if it satisfie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in K if and only if its antipode, - x also lies in K. Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on \R^n. Important examples of convex bodies are the Euclidean ball, the hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ... and the cross-polytope. See also * * References * {{Authority control Multi-dimensional geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugo Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland.. He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern. Mathematical concepts named after Hadwiger Hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in ''d''-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the area, the perimeter, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Metric
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book ''Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice Fréchet in 1906, in his study of the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image of a subset A under the function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system. Often, vertical translations a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation (mathematics)
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire -dimensional flat of fixed points in a -dimensional space. Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Content
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets. It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure. Definition For A \subset \mathbb^, and each integer ''m'' with 0 \leq m \leq n, the ''m''-dimensional upper Minkowski content is :M^(A) = \limsup_ \frac and the ''m''-dimensional lower Minkowski content is defined as :M_*^m(A) = \liminf_ \frac where \alpha(n-m)r^ is the volume of the (''n''−''m'')-ball of radius r and \mu is an n-dimensional Lebesgue measure. If the upper and lower ''m''-dimensional Minkowski content of ''A'' are equal, then their common v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Width
In geometry, the mean width is a measure of the "size" of a body; see Hadwiger's theorem for more about the available measures of bodies. In n dimensions, one has to consider (n-1)-dimensional hyperplanes perpendicular to a given direction \hat in S^, where S^n is the n-sphere (the surface of a (n+1)-dimensional sphere). The "width" of a body in a given direction \hat is the distance between the closest pair of such planes, such that the body is entirely in between the two hyper planes (the planes only intersect with the boundary of the body). The mean width is the average of this "width" over all \hat in S^. More formally, define a compact body B as being equivalent to set of points in its interior plus the points on the boundary (here, points denote elements of \mathbb^n). The support function of body B is defined as : h_B(n)=\max\ where n is a direction and \langle,\rangle denotes the usual inner product on \mathbb^n. The mean width is then : b(B)=\frac \int_ h_B(\hat)+h_B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and real vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Convex Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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