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Liberman's Lemma
Liberman's lemma is a theorem used in studying intrinsic metric, intrinsic geometry of convex surface. It is named after Joseph Liberman. Formulation If \gamma is a unit-speed minimizing geodesic on the surface of a convex body ''K'' in Euclidean space then for any point ''p'' ∈ ''K'', the function : t\mapsto\operatorname^2\circ\gamma(t)-t^2 is concave. References *Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313. Differential geometry of surfaces Lemmas {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intrinsic Metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second along a path whose arclength is equal to (or very close to) that distance. The distance between two points of a metric space relative to the intrinsic metric is defined as the infimum of the lengths of all paths from the first point to the second. A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space. If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic) then it is called a geodesic metric space or geodesic space. For instance, the Euclidean plane is a geodesic space, with line segments as its geodesics. The Euclidean plane with the origin removed is not geodesic, but is still a length metric space. Definitions Let (M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Surface
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions play an important role in many areas of mathematics. They are especially important in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Liberman
Joseph Liberman (1917 in Henichesk – August 1941 in ) was a Soviet mathematician, a student of Aleksandrov, best known for Liberman's lemma. Biography In 1936 he entered Leningrad State University as one of the winners of the first city Mathematical Olimpiad. He worked in geometry and real analysis. In 1938 he published his first paper. In 1939 he entered graduate school. He served in the army since June 1941. He defended Ph.D. thesis in June 1941. At this time he was already lieutenant of Navy anti-aircraft warfare. In August 1941 he was sent to Tallinn, Estonia where he died. Three geometric papers of Liberman appeared in 1941 and another in 1943. See also *Liberman's lemma Liberman's lemma is a theorem used in studying intrinsic metric, intrinsic geometry of convex surface. It is named after Joseph Liberman. Formulation If \gamma is a unit-speed minimizing geodesic on the surface of a convex body ''K'' in Euclidean ... References * ''Александров А. Д. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a " straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of havi ... [...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. Some authors do not require a non-empty interior, merely that the set is non-empty. 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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ... with the unit balls of Norm (mathematics), norms on \R^n. Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope. Metric space structure Write \mathcal K^n for the set of convex bodies in \mathbb R^n. Then \mathcal K^n is a complete metric spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometry, isometric embedding in Euclidean space. Surfaces naturally arise as Graph of a function, graphs of Function (mathematics), functions of a pair of Variable (mathematics), variables, and sometimes appear in parametric form or as Locus (mathematics), loci associated to Curve#Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
