Theorems in geometry

In geometry, a theorem is a statement or proposition that has been proven to be true based on a set of axioms and previously established theorems. Theorems are fundamental to the study of geometry as they provide essential insights and conclusions about geometric figures, relationships, and properties. Theorems in geometry often involve concepts such as points, lines, angles, shapes, and their properties.

Geometric inequalities are mathematical statements that establish relationships between different geometric quantities, such as lengths, areas, angles, and volumes. These inequalities often provide useful bounds or constraints on these quantities and can be applied in various fields, including geometry, optimization, and analysis. Some common types of geometric inequalities include: 1. **Triangle Inequalities**: In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

The triangle inequalities are fundamental properties of triangles related to the lengths of their sides. They state that, for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold: 1. \(a + b > c\) 2. \(a + c > b\) 3.

Berger's isoembolic inequality is a result in the field of differential geometry, particularly concerning Riemannian manifolds. The inequality deals with the comparison of volumes of geodesic balls (or "volumes" in a more general sense) in Riemannian manifolds that have certain curvature bounds.

The Besicovitch inequality is a result in mathematical analysis, particularly in the field of geometric measure theory and harmonic analysis. It is named after the mathematician Aleksandr Besicovitch. The inequality deals with the behavior of measurable functions and their integrals over certain types of sets. One formulation of the Besicovitch inequality can be described for functions defined on a Euclidean space.

The Bishop–Gromov inequality is a fundamental result in Riemannian geometry that provides a comparison between the volume of geodesic balls in a Riemannian manifold and the volume of balls in a model space of constant curvature, specifically spherical or Euclidean spaces. The inequality is particularly useful in the context of manifolds with bounded sectional curvature.

The Blaschke–Lebesgue theorem is a result in the field of measure theory and functional analysis, particularly concerning the properties of certain types of functions in the context of completeness and limit points. The theorem specifically addresses the behavior of sequences of orthogonal functions in a Hilbert space.

The Brascamp–Lieb inequality is an important result in the field of functional analysis and geometric measure theory. It provides a powerful estimate for integrals of products of functions that arise in various areas of mathematics, including harmonic analysis and the theory of partial differential equations. ### Statement of the Inequality The Brascamp–Lieb inequality states that for a collection of measurable functions and linear maps, one can obtain an upper bound on the integral of a product of these functions.

The Gaussian correlation inequality is a result concerning the behavior of Gaussian random variables and their correlations. Specifically, it states that if \( X_1 \) and \( X_2 \) are two jointly distributed Gaussian random variables with the same variance, then their correlation satisfies a specific property regarding their joint distribution. Formally, if \( X_1 \) and \( X_2 \) are standard normal random variables (i.e.

Gromov's systolic inequality is a fundamental result in differential geometry concerning the relationship between the volume and the topology of essential manifolds. Specifically, it addresses the concept of the systole of a Riemannian manifold, which is defined as the length of the shortest nontrivial loop (or closed curve) in the manifold.

The Hitchin–Thorpe inequality is a result in the field of differential geometry, particularly in the study of Riemannian manifolds. It provides a relationship between various geometric and topological properties of compact Riemannian manifolds with a specific focus on their curvature.

Ptolemy's inequality is a mathematical statement that relates the lengths of the sides and diagonals of a cyclic quadrilateral. A cyclic quadrilateral is a four-sided figure (quadrilateral) where all vertices lie on the circumference of a single circle.

Pu's inequality is a result in the field of real analysis, particularly concerning measures and integration. It is associated with the properties of measurable functions and the way in which their integrals behave relative to their suprema. Specifically, Pu's inequality provides a bound on the integral of a non-negative measurable function.

The Pólya–Szegő inequality is a result in the field of mathematics, particularly in the area of functional analysis and inequalities. It provides a comparison of certain integral expressions that involve non-negative functions, and it is often used in the context of orthogonal polynomials and convex functions. More specifically, the Pólya–Szegő inequality deals with the integrals of non-negative functions defined on the interval \([0, 1]\).

The Ring Lemma, also known as the Ring Lemma in the context of topological groups, refers to a result in the field of topology and functional analysis, particularly concerning the structure of certain sets in the context of algebraic operations.

Symmetrization methods refer to a class of mathematical techniques used in various fields such as analysis, probability, and geometry to simplify problems by exploiting symmetries. These methods often transform a given object into a more symmetric one, making it easier to study properties, derive estimates, or provide proofs. ### Key Concepts of Symmetrization Methods: 1. **Symmetrization in Mathematics**: This generally involves replacing a non-symmetric object (like a function or a shape) with a symmetric one.

Toponogov's theorem is a result in the field of differential geometry, specifically relating to the geometry of non-Euclidean spaces such as hyperbolic spaces. It provides a condition for comparing triangles in a geodesic space with triangles in Euclidean space.

Theorems about curves cover a vast range of topics in mathematics, particularly in geometry, calculus, and topology. Here are some key theorems and concepts associated with curves: 1. **Fermat's Last Theorem for Curves**: While Fermat's Last Theorem primarily concerns integers, there are generalizations and discussions about elliptic curves in number theory that relate deeply to the properties of curves.

Fenchel's theorem, often referred to in the context of convex analysis, deals with the correspondence between the convex functions and their subgradients. Specifically, it provides a characterization of convex functions through their conjugate functions.

The Fundamental Theorem of Curves is a concept in differential geometry that establishes a relationship between curves in Euclidean space and the properties of their curvature and torsion. While the specific formulation can vary depending on the context, generally, the theorem addresses the representation of a curve based on its intrinsic geometric properties.

The Jordan Curve Theorem is a fundamental result in topology, a branch of mathematics that studies properties of spaces that are preserved under continuous deformations. The theorem states that any simple closed curve in a plane (a curve that does not intersect itself and forms a complete loop) divides the plane into two distinct regions: an "inside" and an "outside.

Newton's theorem, often referred to as the "Newton's theorem on ovals," relates to the properties of an oval, particularly in the context of projective geometry and combinatorial geometry. The theorem essentially states that given a set of points in the plane, if these points are located on a smooth convex curve (an oval), then there exists a certain relationship concerning the tangents, secants, and other lines drawn from these points.

The Pestov–Ionin theorem is a result in the field of mathematical logic that deals with the preservation of certain properties in structures, particularly in the context of countable models. Although it is a specialized topic, the theorem itself is typically discussed within the framework of model theory, which studies the relationships between formal languages and their interpretations (or models).

The Tennis Ball Theorem is a concept from mathematics, specifically in the area of topology and geometry. It states that every point on the surface of a sphere can be connected to any other point on the sphere by a continuous path that lies entirely on the sphere's surface. This is often illustrated using the analogy of a tennis ball, which is a spherical object.

In complex geometry, theorems often pertain to the study of complex manifolds, complex structures, and the rich interplay between algebraic geometry and differential geometry. Here are some important theorems and concepts in complex geometry: 1. **Kodaira Embedding Theorem**: This theorem states that a compact Kähler manifold can be embedded into projective space if it has enough sections of its canonical line bundle. It is a crucial result linking algebraic geometry with complex manifolds.

The AF + BG theorem is a concept in the field of mathematics, specifically in the area of set theory and topology. However, the notation AF + BG does not correspond to a widely recognized theorem or principle within standard mathematical literature or education. It's possible that this notation is specific to a certain context, course, or area of research that is not broadly covered.

The Appell–Humbert theorem is a result in the theory of complex numbers and multidimensional analysis. It relates to the behavior of certain classes of functions, particularly those that are harmonic or analytic. The theorem states conditions for when a function can be expressed as a series of its values on a certain domain.

The Birkhoff–Grothendieck theorem is a fundamental result in the field of lattice theory and universal algebra. It characterizes the representability of certain types of categories, especially in the context of complete lattice structures. **Statement of the theorem:** The Birkhoff–Grothendieck theorem states that a distributive lattice can be represented as the lattice of open sets of some topological space if and only if it is generated by its finitely generated ideals.

The Bogomolov–Sommese vanishing theorem is a result in algebraic geometry that deals with the vanishing of certain cohomology groups associated with ample line bundles on compact Kähler manifolds.

Hurwitz's automorphisms theorem is a result in the field of group theory and topology, particularly in the study of Riemann surfaces and algebraic curves. It deals with the automorphisms of compact Riemann surfaces and their relationship to the structure of these surfaces.

The Kodaira embedding theorem is a fundamental result in complex differential geometry that provides a criterion for when a compact complex manifold can be embedded into projective space as a complex projective variety. The theorem tackles the interplay between the geometry of a compact complex manifold and the algebraic properties of holomorphic line bundles over it.

Le Potier's vanishing theorem is a result in algebraic geometry concerning sheaf cohomology on certain types of varieties, specifically on smooth projective varieties. It is particularly concerned with the behavior of cohomology groups of coherent sheaves under the action of the derived category.

The Oka coherence theorem is a result in complex analysis and several complex variables, particularly in the field of Oka theory. Named after Shinsuke Oka, this theorem deals with the properties of holomorphic functions and their extensions in certain types of domains.

The Torelli theorem is a fundamental result in algebraic geometry and the theory of Riemann surfaces, attributed to the mathematician Carlo Alberto Torelli. It essentially describes the relationship between the algebraic structure of a curve and its deformation in terms of its Jacobian.

Theorems in plane geometry are propositions or statements that can be proven based on axioms, definitions, and previously established theorems. Plane geometry deals with flat, two-dimensional surfaces and includes the study of points, lines, angles, shapes (such as triangles, quadrilaterals, and circles), and their properties.

Circles are fundamental shapes in geometry, and several important theorems govern their properties and behaviors. Here are some key theorems about circles: 1. **Circumference Theorem**: The circumference \( C \) of a circle is given by the formula: \[ C = 2\pi r \] where \( r \) is the radius of the circle.

Theorems about polygons constitute a significant part of geometry, focusing on the properties, relationships, and characteristics of various types of polygons.

Barbier's theorem is a result in geometry concerning the relationship between the perimeter of a plane figure and the circumference of a circle that has the same area as that figure. Specifically, Barbier's theorem states that for any plane figure, the perimeter of the figure is greater than or equal to the circumference of the circle that has the same area. The equality holds if and only if the figure is a circle.

Hjelmslev's theorem is a result in the field of projective geometry that relates to the properties of conics (i.e., curves defined by quadratic equations) in projective spaces. Specifically, it addresses the conditions under which a conic in one projective plane can be transformed into an equivalent conic in another projective plane.

Holditch's theorem is a result in the field of geometry, specifically in topology related to convex polyhedra. It states that any two convex polyhedra with the same number of vertices, edges, and faces are combinatorially equivalent, meaning they can be transformed into one another through a series of edge-edge and face-face correspondences while preserving the connectivity structure.

The Midpoint Theorem in the context of conics, specifically concerning ellipses, refers to a property related to the midpoints of line segments connecting points on the ellipse. While the term "Midpoint Theorem" can also be associated with other geometrical contexts, such as triangles, in the realm of conics, it is often used to describe certain relationships and properties referring to the midpoints of chords.

The Mohr–Mascheroni theorem is a result in geometry that states that it is possible to construct any length using only a compass, without the need for a straightedge. This theorem is named after the German mathematician Max Mohr and the Italian mathematician Giovanni Mascheroni, who independently proved this result. The theorem can be surprising because traditional geometric constructions often rely on both a compass and a straightedge.

In projective geometry, theorems and principles focus on properties of geometric figures that remain invariant under projective transformations. Projective geometry is primarily concerned with relationships and properties that are not dependent on measurements of distance or angles, but rather on incidence, collinearity, and concurrency.

The Cayley–Bacharach theorem is a result in algebraic geometry that deals with the intersection of divisors on a projective space. It is particularly relevant in the study of linear systems of divisors and their properties. In its classical form, the theorem states the following: Let \( C \) be a non-singular irreducible curve of degree \( d \) in the projective plane \( \mathbb{P}^2 \).

The statement "five points determine a conic" refers to a fundamental result in projective geometry. It states that given any five points in a plane, no three of which are collinear, there exists a unique conic section (which can be an ellipse, parabola, hyperbola, or degenerate conic) that passes through all five points.

Hesse's theorem is a result in geometry that deals with the properties of projective spaces. Specifically, it states that if you have a configuration of points in a projective plane, under certain conditions, the points will lie on a conic (a curve defined by a quadratic polynomial). In a more precise sense, the theorem can be framed in terms of the collinearity of points and the conditions under which these points create a conic.

A Steiner conic, also known as a Steiner curve or a Steiner ellipse, is a specific type of conic section used in projective geometry and other areas of mathematics. It is defined in the context of a given triangle. For a triangle with vertices \( A \), \( B \), and \( C \), the Steiner conic is the unique conic that passes through the triangle's vertices and has the following additional properties: 1. Its foci are located at the triangle's centroid.

The Veblen–Young theorem is a result in set theory and topology that pertains to the structure of certain well-ordered sets and their properties. It is primarily focused on the relationship between well-ordered sets and their representations as ordinals, specifically in the context of a well-ordered set being isomorphic to an ordinal if it exhibits certain properties.

The term "2π theorem" doesn't refer to a widely recognized theorem in mathematics or physics by that name. However, it might be associated with concepts involving the number \(2\pi\), which frequently appears in contexts related to circles, trigonometry, and periodic functions.

The Almgren regularity theorem is a result in the field of geometric measure theory, specifically concerning the regularity properties of minimizers of certain variational problems. Named after the mathematician Frederic J. Almgren Jr., the theorem addresses the behavior of minimizers of the area functional, which are often studied in the context of minimal surfaces.

Anderson's theorem, formulated by P.W. Anderson in the context of condensed matter physics, primarily relates to the behavior of disordered systems, particularly in the study of superconductivity and localization effects. The theorem is often associated with the concept of Anderson localization, which describes how wavefunctions (such as those of electrons) can become localized in a disordered medium and thus inhibit electrical conductivity.

Bang's theorem on tetrahedra is a result in geometry regarding the arrangement of points within a tetrahedron. Specifically, it concerns the maximal number of points that can be placed in the interior of a tetrahedron such that no three points are coplanar.

Blichfeldt's theorem is a result in the field of number theory, specifically in the study of lattice points and their distributions. Named after the mathematician A.B. Blichfeldt, the theorem deals with the packing of points in a convex geometry context.

Campbell's theorem is a result in differential geometry that pertains to the geometry of a Euclidean space and the properties of certain curves and surfaces within it. Specifically, it deals with the concept of the Frenet frame and the curvature of curves. In its essence, Campbell's theorem states that for a certain class of curves in Euclidean space, there exists a correspondence between curvature and torsion.

Castelnuovo's contraction theorem is a result in algebraic geometry, specifically dealing with the properties of smooth projective varieties. The theorem is part of the study of the behavior of certain types of morphisms between algebraic varieties, particularly in the context of contraction maps in the context of the minimal model program (MMP).

The Castelnuovo–de Franchis theorem is a result in algebraic geometry that deals with the embedding of algebraic curves in projective space. More specifically, it provides conditions under which a non-singular projective curve can be embedded into projective space of a certain dimension based on its genus and the degree of a line bundle.

The Collage Theorem, often referred to in the context of topology and geometry, is a concept related to the study of spaces and continuous functions. However, the term "Collage Theorem" may not be universally recognized under that name in all areas of mathematics, and its interpretation can vary depending on the context.

The Double Limit Theorem, often referred to in the context of limits in calculus, relates to the properties and behavior of limits involving functions of two variables.

Dévissage is a French term that translates to "unscrewing" in English. In various contexts, it can refer to the act of removing screws or bolts from an object. However, the term can also have specialized meanings in different fields. In the context of watchmaking, for example, dévissage refers to the process of unscrewing the crown of a watch to adjust the time or date.

Euler's rotation theorem states that any rotation of a rigid body in three-dimensional space can be represented as a single rotation about a specific axis. This means that for any arbitrary rotation, it is possible to find an axis in space such that the body can be considered to have rotated around this axis by a specific angle. More formally, the theorem states that given any rotation defined by a rigid body transformation, there exists a unique axis of rotation and a corresponding angle of rotation about that axis.

The Fold-and-Cut theorem is a result in computational geometry and combinatorial geometry that deals with the problem of folding paper to achieve a desired cut. Specifically, it states that any shape that can be formed by a straight cut through a folded piece of paper can be realized by an appropriate folding of the paper beforehand.

The Hyperbolization Theorem is a result in the field of topology and geometric group theory, specifically concerning the characteristics of 3-manifolds. It states that any compact, orientable 3-manifold that contains a certain type of submanifold (specifically, a “reducible” submanifold or one that can be "hyperbolized") can be decomposed into pieces that exhibit hyperbolic geometry.

Jørgensen's inequality is a result in the field of functional analysis, particularly concerning the relationships between norms in Banach spaces. Specifically, Jørgensen's inequality pertains to the estimates of certain linear operators and is often discussed in the context of submartingales, Brownian motion, and processes in probability theory.

Lexell's theorem, often associated with the field of celestial mechanics, pertains to the motion of celestial bodies in gravitational fields. Specifically, it describes the precession or gradual change in the orientation of the orbit of a celestial body due to perturbations from other bodies or non-uniformities in the gravitational field.

The Lickorish–Wallace theorem is a result in the field of topology, specifically in the study of 3-manifolds. This theorem provides a criterion for when a connected sum of 3-manifolds can be represented as a connected sum of prime 3-manifolds.

Liouville's theorem in the context of conformal mappings relates to the properties of holomorphic (or analytic) functions defined on the complex plane. Specifically, the theorem states that any entire (holomorphic everywhere in the complex plane) function that is bounded is constant.

The Non-Squeezing Theorem is a fundamental result in symplectic geometry, a branch of mathematics that studies structures and properties of spaces that are equipped with a symplectic form. Specifically, the theorem addresses the concept of symplectic embeddings, which are mappings between symplectic manifolds that preserve the symplectic structure. The Non-Squeezing Theorem asserts that there are limitations on how one can "squeeze" or transform symplectic spaces.

Pappus's centroid theorem, named after the ancient Greek mathematician Pappus of Alexandria, is a principle concerning the geometry of figures in relation to their centroids (or centroids). It actually consists of two related theorems, often referred to as Pappus's centroid theorems.

The Petersen–Morley theorem is a result in graph theory that concerns the structure of certain types of graphs. It states that for every sufficiently large graph, if it contains no complete subgraph \( K_n \) of size \( n \), then the graph can be colored with \( n-1 \) colors such that no two adjacent vertices share the same color. The theorem is particularly relevant when discussing the properties of planar graphs and colorability.

The Riemannian Penrose inequality is a result in differential geometry and general relativity that relates the total mass of a Riemannian manifold with boundary to the area of its boundary. It is an extension of the classical Penrose inequality, which is a key result in the theory of general relativity regarding the mass of gravitational systems.

The Skoda–El Mir theorem is a result in complex analysis, specifically in the theory of several complex variables and the study of holomorphic functions. It pertains to the properties of holomorphic functions defined on complex manifolds, particularly focusing on the behavior of such functions near their zero sets. In essence, the theorem addresses the relationships between the zero sets of holomorphic functions and their implications for the analyticity and continuity of these functions.

Soddy's hexlet is a configuration in geometry involving three circles that are tangent to each other in a specific way. Named after the British chemist Frederick Soddy, who explored this arrangement in connection with the theory of circles, Soddy's hexlet refers to the construction of two smaller circles that are tangent to three larger circles, along with two additional larger circles that touch the three originals.

The Spherical Law of Cosines is a fundamental theorem in spherical geometry, which deals with the relationships between the angles and sides of spherical triangles (triangles drawn on the surface of a sphere). Specifically, it is used to relate the lengths of the sides of a spherical triangle and the cosine of one of its angles.

The term "Theorem of the cube" is not widely recognized in mathematics as a specific theorem. However, it could refer to various concepts depending on the context.

The Thom conjecture, proposed by mathematician René Thom in the 1950s, relates to topology and singularity theory. Specifically, it concerns the structure of non-singular mappings between manifolds and the conditions under which certain types of singularities can occur. The conjecture asserts that every real-valued function defined on a manifold can be approximated by a function that has a certain type of "generic" singularity.

The term "Ultraparallel theorem" is not widely recognized in established mathematical literature or common mathematical terminology. However, it is possible that you are referring to a theorem related to non-Euclidean geometries or the properties of parallel lines. In the context of hyperbolic geometry, for example, two lines may be defined as "ultraparallel" if they do not intersect and are not parallel in the sense used in Euclidean geometry.