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Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to
Minkowski's_theorem
Theorem in geometry
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures)
Brunn–Minkowski_theorem
volume in an appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality
Minkowski–Steiner_formula
Application of geometry in number theory
{\displaystyle K} is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if vol ( K ) > 2 n vol ( R
Geometry_of_numbers
Theorem in algebraic number theory
In mathematics, especially in algebraic number theory, the Hermite–Minkowski theorem states that for any integer N there are only finitely many number
Hermite–Minkowski_theorem
Theorem in geometric number theory
In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental
Minkowski's_second_theorem
Limits ideals to be checked in order to determine the class number of a number field
easy to show that the lower bound is greater than 1, so we obtain Minkowski's Theorem, that the discriminant of every number field, other than Q, is non-trivial
Minkowski's_bound
German mathematician and physicist (1864–1909)
Hermann Minkowski Abraham–Minkowski controversy Brunn–Minkowski theorem Hasse–Minkowski theorem Hermite–Minkowski theorem Minkowski addition Minkowski (crater)
Hermann_Minkowski
Existence theorem on the lattice packing of hyperspheres
In mathematics, the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in
Minkowski–Hlawka_theorem
High-area shapes can shift to hold many grid points
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area A {\displaystyle
Blichfeldt's_theorem
Local–global principle for quadratic forms
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only
Hasse–Minkowski_theorem
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Formula for area of a grid polygon
these triangles have area 1 2 {\displaystyle {\tfrac {1}{2}}} uses Minkowski's theorem that a symmetric convex set centered at a grid point and with no
Pick's_theorem
Sums vector sets A and B by adding each vector in A to each vector in B
fundamental in the Lp Brunn-Minkowski theory. Blaschke sum – Polytope combining two smaller polytopes Brunn–Minkowski theorem – Theorem in geometry, an inequality
Minkowski_addition
Surname list
diagram Minkowski distance Minkowski functional Minkowski inequality Minkowski space Null vector (Minkowski space) Minkowski plane Minkowski's theorem Minkowski's
Minkowski
Concept in number theory
And we proved the theorem. Another simple proof of the Dirichlet's approximation theorem is based on Minkowski's theorem applied to the set S = {
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Measures the size of the ring of integers of the algebraic number field
Kronecker first stated Minkowski's theorem in 1882, though the first proof was given by Hermann Minkowski in 1891. In the same year, Minkowski published his bound
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
theorem (logic) Diaconescu's theorem (mathematical logic) Easton's theorem (set theory) Erdős–Dushnik–Miller theorem (set theory) Erdős–Rado theorem (set
List_of_theorems
the Minkowski problem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets. The theorem that
Minkowski problem for polytopes
Minkowski_problem_for_polytopes
Natural number
04284 [math.NT]. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate
33_(number)
Mathematical concept
reduced the number field case to the function field case. The Hasse–Minkowski theorem is a fundamental result in number theory that states that two quadratic
Global_field
Nonexistence result for combinatorial block designs
Bruck–Ryser–Chowla theorem are not merely necessary, but also sufficient for the existence of such a rational matrix R. They can be derived from the Hasse–Minkowski theorem
Bruck–Ryser–Chowla_theorem
On the existence of hyperplanes separating disjoint convex sets
are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector
Hyperplane_separation_theorem
Minkowski (1864 - 1909), German mathematician: Brunn–Minkowski theorem Hasse–Minkowski theorem Hermite–Minkowski theorem Minkowski addition Minkowski
List of things named after Hermann Minkowski
List_of_things_named_after_Hermann_Minkowski
Indefinite quadratic forms in > 4 variables over the rationals nontrivially represent 0
integral solution x may also be found. Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement:
Meyer's_theorem
Mathematical description of spacetime used in relativity
representation theorem, may be expressed as the action of a linear functional on the vector space, the same holds for the Minkowski inner product of Minkowski spacetime
Minkowski_spacetime
Natural number
Retrieved 2016-05-31. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate
29_(number)
Solving integer equations from all modular solutions
the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer. On the other hand, Birch's theorem shows that
Hasse_principle
Israeli-American computer scientist
ISSN 0302-9743. Regev, Oded; Stephens-Davidowitz, Noah (2017), A reverse Minkowski theorem, Annual ACM SIGACT Symposium on Theory of Computing, Montreal, Quebec
Oded Regev (computer scientist)
Oded_Regev_(computer_scientist)
Topics referred to by the same term
theorem Hasse's theorem on elliptic curves Hasse–Arf theorem Hasse–Minkowski theorem Hasse principle, the principle that an integer equation can be solved
Hasse's_theorem
Upper bound on the volume of a convex body containing one lattice point
only one lattice point in its interior. It is a kind of converse to Minkowski's theorem, which guarantees that a centrally symmetric convex body K {\displaystyle
Ehrhart's_volume_conjecture
Relation between sides of a right triangle
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle
Pythagorean_theorem
Natural number
Retrieved 2020-08-07. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate
7
Overview of and topical guide to geometry
Homothetic center Hyperplane Lattice Ehrhart polynomial Leech lattice Minkowski's theorem Packing Sphere packing Kepler conjecture Kissing number problem Honeycomb
Outline_of_geometry
Natural number
Retrieved 2023-10-09. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate
63_(number)
Hajós's theorem was simplified by Tibor Szele. An equivalent statement on homogeneous linear forms was originally conjectured by Hermann Minkowski. A consequence
Hajós's_theorem
Every Riemannian manifold can be isometrically embedded into some Euclidean space
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded
Nash_embedding_theorems
Natural number
Retrieved 2023-10-09. Cohen, Henri (2007). "Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Graduate
21_(number)
principle Hasse–Minkowski theorem Galois module Galois cohomology Brauer group Class field theory Abelian extension Kronecker–Weber theorem Hilbert class
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
non-singular quadratic form of dimension 2 or more is universal. The Hasse–Minkowski theorem implies that a form is universal over Q {\displaystyle \mathbb {Q}
Universal_quadratic_form
Polytope combining two smaller polytopes
the Kneser–Süss inequality, an analogue of the Brunn–Minkowski theorem on volumes of Minkowski sums of convex bodies: V ( X # Y ) ( d − 1 ) / d ≥ V (
Blaschke_sum
In mathematics, invariant of square matrices
) . {\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}} Brunn–Minkowski theorem implies that the nth root of determinant is a concave function, when
Determinant
On the approximate structure of sets whose sumset is small
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose
Freiman's_theorem
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
Type of Benz planes
hyperbolas.) Theorem (Chen): Only a Minkowski plane M ( K ) {\displaystyle {\mathfrak {M}}(K)} satisfies the theorem of Miquel. Because of the last theorem M (
Minkowski_plane
Book on the geometry of numbers
results in Diophantine approximation. The chapters on Minkowski's theorem and Blichfeldt's theorem, particularly, have been called the "foundation stones"
The_Geometry_of_Numbers
of numbers Minkowski's theorem Pick's theorem Mahler's compactness theorem Mahler measure Effective results in number theory Mahler's theorem Brun sieve
List_of_number_theory_topics
Non-empty convex set in Euclidean space
containing, or contained in, an n-dimensional convex object Brunn–Minkowski theorem, which has many implications relevant to the geometry of convex bodies
Convex_body
Classify quadratic forms over algebraic number fields
He published his work in 1923 and 1924. See Hasse principle, Hasse–Minkowski theorem. The local-global principle says that a general result about a rational
Hilbert's_eleventh_problem
Shape designed to roll down some path
repeating the series twice. The existence of trajectoids follows from Minkowski's theorem and the Rodrigues formula for the product of rotation matrices. Sobolev
Trajectoid
Smooth 4-manifold homeomorphic yet not diffeomorphic to Euclidean space
-E_{8}\oplus [+1]\oplus [-1]} with the isomorphism coming from the Hesse–Minkowski theorem. It is therefore the same as for the topological 4-manifold M E 8
Exotic_R4
Theorem in projective geometry
In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem, Latin for mystical hexagram) states that if six arbitrary points
Pascal's_theorem
Hungarian and American mathematician and physicist (1903–1957)
of calculus of variations, and a small simplification of Hermann Minkowski's theorem for linear forms in geometric number theory. Later in his career
John_von_Neumann
Branch of number theory
an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many integral ideals with norm less
Algebraic_number_theory
Theorem in physics showing the conservation of energy for the electromagnetic field
In electrodynamics, Poynting's theorem is a statement of conservation of energy for electromagnetic fields that was developed by British physicist John
Poynting's_theorem
All curves of constant width have the same perimeter
answer. One proof of the theorem uses the properties of Minkowski sums. If K is a body of constant width w, then the Minkowski sum of K and its 180° rotation
Barbier's_theorem
Computationally feasible measure in real-valued number of dimensions
in higher dimensions Minkowski–Bouligand dimension Federer 1969, p. 273 Krantz & Parks 1999, p. 74 Federer 1969, p. 275, Theorem 3.2.39 Federer, Herbert
Minkowski_content
In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed. Let X {\displaystyle
Dieudonné's_theorem
Number-theoretical function
1080/00150517.1993.12429300. Cohen, H. (2007). "5.4 Consequences of the Hasse–Minkowski Theorem". Number Theory Volume I: Tools and Diophantine Equations. Springer
Sum_of_squares_function
in S⊥. Brunn–Minkowski inequality Prékopa–Leindler inequality Busemann, Herbert (1949). "A theorem on convex bodies of the Brunn-Minkowski type". Proc
Busemann's_theorem
Constant relating to close packing of spheres
{\displaystyle \Gamma (x)} is the gamma function. Loewner's torus inequality Minkowski's theorem Cassels (1971) p. 36 Kitaoka (1993) p. 36 Blichfeldt, H. F. (1929)
Hermite_constant
Algebraic topology Discrete space Fundamental group Geometry Homology Minkowski's theorem Topological group Field Finite field Galois theory Grothendieck group
List_of_group_theory_topics
Key result in general relativity
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential
Positive_energy_theorem
On when a function on convex body K does not decrease if K is translated inwards
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative
Anderson's_theorem
American-born British mathematician (1888-1972)
Manchester Literary & Philosophical Society. Mordell, L. J. (1937). "Minkowski's theorems and hypotheses on linear forms". Comptes rendus du Congrès international
Louis_J._Mordell
Theorem on extension of bounded linear functionals
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace
Hahn–Banach_theorem
Triangle inequality in Lp spaces
Littlewood & Pólya 1988, Theorem 202. Bahouri, Chemin & Danchin 2011, p. 4. Mulholland, H. P. (1949). "On Generalizations of Minkowski's Inequality in the Form
Minkowski_inequality
Algebraic structure with addition, multiplication, and division
technique is called the local–global principle. For example, the Hasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic equations
Field_(mathematics)
German mathematician (1898–1979)
Kurt Hensel, writing a dissertation in 1921 containing the Hasse–Minkowski theorem, as it is now called, on quadratic forms over number fields. He then
Helmut_Hasse
/}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})} given by Minkowski's theorem applied to Euclidean balls. This family comes with a polynomial time
Barnes–Wall_lattice
Law Field Person(s) Named After Abel's theorem Calculus Niels Henrik Abel Ariadne's thread Computer science Ariadne Amdahl's law Computer science Gene
List of scientific laws named after people
List_of_scientific_laws_named_after_people
Theorem in axiomatic quantum field theory
Reeh–Schlieder theorem is a result in relativistic local quantum field theory published by Helmut Reeh and Siegfried Schlieder in 1961. The theorem states that
Reeh–Schlieder_theorem
Mathematical theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of
Riesz–Fischer_theorem
German mathematician (1905–1988)
minimization Fenchel's duality theorem Geometry Convex geometry Brunn–Minkowski theorem Differential geometry Fenchel's theorem Hyperbolic geometry Jakob Nielsen
Werner_Fenchel
Theorem in integral geometry
W_{n-j}.} Minkowski functional – Function made from a set Set function – Function from sets to numbers An account and a proof of Hadwiger's theorem may be
Hadwiger's_theorem
Polynomial with all terms of degree two
of a quadratic form Hasse–Minkowski theorem Quadric Ramanujan's ternary quadratic form Square class Witt group Witt's theorem A tradition going back to
Quadratic_form
Uniqueness of countable dense linear orders
Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named after Georg
Cantor's_isomorphism_theorem
Function made from a set
on K . {\textstyle K.} Theorem—If K {\textstyle K} is an absorbing disk in a vector space X {\textstyle X} then the Minkowski functional of K , {\textstyle
Minkowski_functional
Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets
Vitale's random Brunn–Minkowski inequality
Vitale's_random_Brunn–Minkowski_inequality
invariants and the signatures coming from real embeddings. Hasse–Minkowski theorem Lam (2005) p.118 Milnor & Husemoller (1973) p.79 Serre (1973) p.36
Hasse invariant of a quadratic form
Hasse_invariant_of_a_quadratic_form
On when a space equals the closed convex hull of its extreme points
Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex
Krein–Milman_theorem
Statement on the gravitational attraction of spherical bodies
shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetric body. This theorem has particular
Shell_theorem
polynomial Hermite–Kronecker–Brioschi characterization The Hermite–Minkowski theorem, stating that only finitely many number fields have small discriminants
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Theorem in projective geometry
In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: Any oval in a finite pappian projective
Segre's_theorem
Black holes are characterized only by mass, charge, and spin
The no-hair theorem, also known as the black hole uniqueness theorem, states that all stationary black hole solutions of the Einstein–Maxwell equations
No-hair_theorem
Geometry problem on tiling by hypercubes
Minkowski was led to a special case of the cube-tiling conjecture from a problem in diophantine approximation. One consequence of Minkowski's theorem
Keller's_conjecture
\mathbb {Z} } or Z for the set of integer numbers. For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
On the number of common zeros of Laurent polynomials
The Bernstein–Kushnirenko theorem, also called Bernstein–Khovanskii–Kushnirenko theorem (BKK theorem), states that the number of nonzero complex solutions
Bernstein–Kushnirenko_theorem
Unit-distance-preserving maps are isometries
In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit
Beckman–Quarles_theorem
Periodic set of points
d ( Λ ) = 1 {\displaystyle d(\Lambda )=1} is called unimodular. Minkowski's theorem relates the number d ( Λ ) {\displaystyle \mathrm {d} (\Lambda
Lattice_(group)
Theorem in theoretical physics
In theoretical physics, the Haag–Łopuszański–Sohnius theorem states that if both commutating and anticommutating generators are considered, then the only
Haag–Łopuszański–Sohnius theorem
Haag–Łopuszański–Sohnius_theorem
American mathematician (1923–2004)
J. (1961). "Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem". Canadian Journal of Mathematics. 13: 444–453. doi:10.4153/CJM-1961-037-0
William_J._Firey
Construct all metric spaces where lines resemble those on a sphere
summarizes as follows: The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the
Hilbert's_fourth_problem
Key results in general relativity on gravitational singularities
The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the
Penrose–Hawking singularity theorems
Penrose–Hawking_singularity_theorems
Sums of sets of vectors are nearly convex
For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper
Shapley–Folkman_lemma
Mathematical model of the physical space
intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel
Euclidean_geometry
Theorem in general relativity
Lovelock's theorem of general relativity says that from a local gravitational action which contains only second derivatives of the four-dimensional spacetime
Lovelock's_theorem
Mathematical problem
additional restrictions on the attainability of the bound using the Hasse–Minkowski theorem on the rational equivalence of quadratic forms, and showed that the
Hadamard's maximal determinant problem
Hadamard's_maximal_determinant_problem
Constraints on possible particle properties
In theoretical physics, the Weinberg–Witten (WW) theorem, proved by Steven Weinberg and Edward Witten, states that massless particles (either composite
Weinberg–Witten_theorem
Soviet mathematician (1899–1981)
Lusternik–Schnirelmann category Lyusternik's generalization of the Brunn–Minkowski theorem Pavel Alexandrov et al., LAZAR' ARONOVICH LYUSTERNIK (on the occasion
Lazar_Lyusternik
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