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Triangle inequality in Lp spaces
mathematical analysis, the Minkowski inequality establishes that the L p {\displaystyle L^{p}} spaces satisfy the triangle inequality in the definition of normed
Minkowski_inequality
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
Lieb–Thirring inequality Littlewood's 4/3 inequality Markov brothers' inequality Mashreghi–Ransford inequality Max–min inequality Minkowski's inequality Poincaré
List_of_inequalities
Integral inequality
Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of
Prékopa–Leindler_inequality
mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely
Minkowski's first inequality for convex bodies
Minkowski's_first_inequality_for_convex_bodies
reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex
Milman's reverse Brunn–Minkowski inequality
Milman's_reverse_Brunn–Minkowski_inequality
Function spaces generalizing finite-dimensional p norm spaces
\mathrm {d} \mu .} Minkowski inequality, which states that ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} satisfies the triangle inequality, can be generalized:
Lp_space
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
Mathematical description of spacetime used in relativity
In physics, Minkowski spacetime (or Minkowski space; /mɪŋˈkɔːfski, -ˈkɒf-/) is the main mathematical description of spacetime in the absence of gravitation
Minkowski_spacetime
German mathematician and physicist (1864–1909)
Minkowski (crater) Minkowski distance Minkowski functional Minkowski inequality Minkowski model Minkowski plane Minkowski problem Minkowski problem for polytopes
Hermann_Minkowski
M. (1984). "On the similarity of the entropy-power inequality and the Brunn-Minkowski inequality". IEEE Trans. Inf. Theory. 30 (6): 837–839. doi:10.1109/TIT
Entropy_power_inequality
Inequality between integrals in Lp spaces
{\displaystyle L^{1}(\mu )} . Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p ( μ ) {\displaystyle
Hölder's_inequality
Sums vector sets A and B by adding each vector in A to each vector in B
Lp Brunn-Minkowski theory. Blaschke sum – Polytope combining two smaller polytopes Brunn–Minkowski theorem – Theorem in geometry, an inequality on the volumes
Minkowski_addition
Mathematical inequality relating inner products and norms
Kunita–Watanabe inequality Lagrange's identity – On products on sums of squares Minkowski inequality – Triangle inequality in Lp spaces Paley–Zygmund inequality – Probability
Cauchy–Schwarz_inequality
Geometric inequality applicable to any closed curve
the "corona" may be a curve. The proof of the inequality follows directly from Brunn–Minkowski inequality between a set S {\displaystyle S} and a ball
Isoperimetric_inequality
Property of geometry, also used to generalize the notion of "distance" in metric spaces
But it does not hold in some contexts, for example Minkowski spacetime. The triangle inequality is one of the defining properties of a distance function
Triangle_inequality
Mathematical relation making a non-equal comparison
Markov's inequality Minkowski inequality Nesbitt's inequality Pedoe's inequality Poincaré inequality Samuelson's inequality Sobolev inequality Triangle
Inequality_(mathematics)
Minkowski content Minkowski distance Minkowski functional Minkowski inequality Minkowski model Minkowski plane Minkowski problem Minkowski problem for polytopes
List of things named after Hermann Minkowski
List_of_things_named_after_Hermann_Minkowski
Vector distance function
_{i=1}^{n}|x_{i}|^{p}\right)^{1/p}} The Minkowski distance is a metric as a result of the Minkowski inequality, ‖ X + Y ‖ p ≤ ‖ X ‖ p + ‖ Y ‖ p . {\displaystyle
Minkowski_distance
Surname list
diagram Minkowski distance Minkowski functional Minkowski inequality Minkowski space Null vector (Minkowski space) Minkowski plane Minkowski's theorem
Minkowski
Study of geometric properties of sets through measure theory
isoperimetric inequality. The Brunn–Minkowski inequality also leads to Anderson's theorem in statistics. The proof of the Brunn–Minkowski inequality predates
Geometric_measure_theory
Mathematical inequality about the convolution of two functions
and g {\displaystyle g} are multidimensional Gaussian functions. Minkowski inequality Young, W. H. (1912), "On the multiplication of successions of Fourier
Young's convolution inequality
Young's_convolution_inequality
appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after
Minkowski–Steiner_formula
Geometric inequality or concentration inequality in mathematics and probability theory
PMC 4847755. PMID 27134693. Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bulletin of the American Mathematical Society. New Series
Brascamp–Lieb_inequality
Average value of a random variable
of p = q = 2 is called the Cauchy–Schwarz inequality, and is particularly well-known. Minkowski inequality: given any number p ≥ 1, for any random variables
Expected_value
Operation in calculus
dx\right)^{1/q}.} For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality. Minkowski inequality. Suppose that p ≥ 1 is a real number and
Integral
Theorem in mathematics
doi:10.1007/s002220100160. Gardner, Richard J. (2002). "The Brunn–Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic)
Borell–Brascamp–Lieb inequality
Borell–Brascamp–Lieb_inequality
Notion in statistics
like the Minkowski-Steiner formula. The remainder of the proof uses the entropy power inequality, which is like the Brunn–Minkowski inequality. The trace
Fisher_information
Computationally feasible measure in real-valued number of dimensions
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
Minkowski_content
\over n}n=1.} Clearing denominators then gives the desired result. Minkowski inequality Minkowski inequality in the Encyclopedia of Mathematics v t e
Mahler's_inequality
Type of metric geometry
Riesz. In developing the geometry of numbers, Hermann Minkowski established his Minkowski inequality, stating that these spaces define normed vector spaces
Taxicab_geometry
Application of geometry in number theory
bodies is the Minkowski sum A + B = { a + b : a ∈ A , b ∈ B } {\displaystyle A+B=\{a+b:a\in A,\ b\in B\}} . The Brunn–Minkowski inequality relates the
Geometry_of_numbers
Way to associate a non-negative number to a tuple of convex bodies
K_{n})}}.} Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality, are special cases of the
Mixed_volume
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
Concept in mathematics
missing publisher (link) Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10
Layer_cake_representation
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
Mathematical theorem
L^{p}.} When 1 ≤ p ≤ ∞ , {\displaystyle 1\leq p\leq \infty ,} the Minkowski inequality implies that the Lp space L p {\displaystyle L^{p}} is a normed space
Riesz–Fischer_theorem
American mathematical physicist
Prékopa-Leindler inequality to other types of convex combinations of two positive functions. He strengthened the inequality and the Brunn-Minkowski inequality by introducing
Elliott_H._Lieb
Method of determining fractal dimension
In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal
Minkowski–Bouligand_dimension
of inequalities Triangle inequality Bernoulli's inequality Cauchy–Schwarz inequality Hölder's inequality Minkowski inequality Jensen's inequality Chebyshev's
List_of_real_analysis_topics
Inequality in mathematics
_{0}^{1}\left(\int _{0}^{\infty }f(sx)^{p}\,dx\right)^{1/p}\,ds} by Minkowski's integral inequality. Finally, by another change of variables, the last expression
Hardy's_inequality
Branch of mathematics
sections away from the median level, a consequence of the Brunn–Minkowski inequality. Under certain general conditions, a function defined on a high-dimensional
Asymptotic_geometry
American mathematician
Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong (2012), "The log-Brunn—Minkowski inequality" (PDF), Advances in Mathematics, 231 (3–4): 1974–1997, doi:10.1016/j
Gaoyong_Zhang
American mathematician
Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong (2012), "The log-Brunn—Minkowski inequality", Advances in Mathematics, 231 (3–4): 1974–1997, doi:10.1016/j.aim
Erwin_Lutwak
Inequality applying to Eudlidean space and half-spaces
ISSN 1573-8795. S2CID 121935322. Borell, Christer (1975). "The Brunn-Minkowski Inequality in Gauss Space". Inventiones Mathematicae. 30 (2): 207–216. Bibcode:1975InMat
Gaussian isoperimetric inequality
Gaussian_isoperimetric_inequality
German mathematician (1862–1939)
mathematician, known for his work in convex geometry (see Brunn–Minkowski inequality) and in knot theory. Brunnian links are named after him, as his 1892
Hermann_Brunn
Mathematical theory
using methods from convex geometry, such as the Brunn-Minkowski inequality, mixed Minkowski volumes and Steiner's formula. A crucial step towards a
Finite_sphere_packing
Fundamental theorem of algebra Lambda calculus Invariance of domain Minkowski inequality Nash embedding theorem Open mapping theorem (functional analysis)
List_of_mathematical_proofs
Non-empty convex set in Euclidean space
4230/LIPIcs.SoCG.2023.9. Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10
Convex_body
Boston, MA: Birkhäuser, MR 1995384 Okounkov, Andrei (1996), "Brunn–Minkowski inequality for multiplicities", Inventiones Mathematicae, 125 (3): 405–411,
Newton–Okounkov_body
N-th root of the arithmetic mean of the given numbers raised to the power n
mean Average Heronian mean Inequality of arithmetic and geometric means Lehmer mean – also a mean related to powers Minkowski distance Quasi-arithmetic
Generalized_mean
Algebraic structure in linear algebra
triangle inequality for ‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p {\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}} is provided by the Minkowski inequality. For technical
Vector_space
Polytope combining two smaller polytopes
{\displaystyle Y} obeys an inequality known as the Kneser–Süss inequality, an analogue of the Brunn–Minkowski theorem on volumes of Minkowski sums of convex bodies:
Blaschke_sum
Sums of sets of vectors are nearly convex
Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively understood
Shapley–Folkman_lemma
Polygon associated with a compact Riemann surface
theory, the geometry of numbers and circle packing, such as the Brunn–Minkowski inequality. Two elementary proofs due to H. S. M. Coxeter and Voronoi will be
Fundamental_polygon
(B)^{1-\lambda },} where λ A + (1 − λ) B denotes the Minkowski sum of λ A and (1 − λ) B. The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave
Logarithmically concave measure
Logarithmically_concave_measure
Ukrainian-Israeli mathematician (born 1939)
analysis. His results in this field include Milman's reverse Brunn–Minkowski inequality and the quotient of subspace theorem. He holds several positions
Vitali_Milman
person.) Hermann Minkowski, German mathematician – Minkowski addition, Minkowski inequality, Minkowski space, Minkowski diagram, Minkowski's theorem. Minos
List_of_eponyms_(L–Z)
On when a function on convex body K does not decrease if K is translated inwards
origin-symmetric convex body K ⊆ Rn. Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10
Anderson's_theorem
Ellipsoid most closely containing, or contained in, an n-dimensional convex object
original (PDF) on 2017-01-16. Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10
John_ellipsoid
In geometry, set whose intersection with every line is a single line segment
Mathematical Inequalities & Applications (2): 301–348. arXiv:1404.6808. doi:10.7153/mia-20-22. ISSN 1331-4343. The empty set is important in Minkowski addition
Convex_set
Generalization of the Legendre transformation
(strict) epigraph of the infimal convolution of two functions is the Minkowski sum of the (strict) epigraphs of those functions. If the function f {\displaystyle
Convex_conjugate
Hungarian mathematician
1007/BF01876039. S2CID 121469006. Ruzsa, Imre Z. (1997). "The Brunn-Minkowski inequality and nonconvex sets". Geometriae Dedicata. 67 (3): 337–348. doi:10
Imre_Z._Ruzsa
Function made from a set
mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion
Minkowski_functional
functions on both points and momenta; not just functions on points. Minkowski Minkowski inequality modulus modulus of continuity. Montel Montel's theorem. monotone
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Integral geometry Random coil Stochastic geometry Vitale's random Brunn–Minkowski inequality Benford's law Pareto principle History of probability Newton–Pepys
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Canadian-American mathematician (1925–2020)
achievements include the resolution of the Minkowski problem in two-dimensions, the Gagliardo–Nirenberg interpolation inequality, the Newlander-Nirenberg theorem
Louis_Nirenberg
Geometric inequality
Bonnesen's inequality is an inequality relating the length, the area, the radius of the incircle and the radius of the circumcircle of a Jordan curve
Bonnesen's_inequality
Non-Euclidean geometry
isometrically embedded inside the ( n + 1 ) {\displaystyle (n+1)} -dimensional Minkowski space (which is not a Riemannian but rather a Lorentzian manifold). More
Hyperbolic_space
Constraints on energy distribution in spacetime
University. Michael J. Pfenning's work on quantum inequalities showed that in a 2-D spacetime (Minkowski and Rindler) , the energy of the electromagnetic
Quantum_inequalities
Length of a line segment
which measures distance as the sum of the distances in each coordinate. Minkowski distance (Lp distance), a generalization that unifies Euclidean distance
Euclidean_distance
Petty 1967. Schneider 1967. Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bulletin of the American Mathematical Society. New Series. 39 (3):
Shephard's_problem
Existence of a line through two points
combinatorial structure closely connected to zonohedra, polyhedra formed as the Minkowski sum of a finite set of line segments, called generators. In this connection
Sylvester–Gallai_theorem
American mathematician (1925–2013)
1090/s0002-9904-1964-11159-9. MR 0162198 Flanders, Harley (1968). "A proof of Minkowski's inequality for convex curves". Amer. Math. Monthly. 75 (6): 581–593. doi:10
Harley_Flanders
Mathematical inequality relating the derivative of a function to its covariant derivative
diamagnetic inequality relates the Sobolev norm of the absolute value of a section of a line bundle to its covariant derivative. The diamagnetic inequality has
Diamagnetic_inequality
Mathematical function
with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm
Seminorm
Overview of and topical guide to geometry
geometry Pseudosphere Tractricoid Elliptic geometry Spherical geometry Minkowski space Thurston's conjecture Parametric curve Bézier curve Spline Hermite
Outline_of_geometry
Chinese-American mathematician (born 1949)
complete. For instance, they showed that if M is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then
Shing-Tung_Yau
Family of linear transformations
a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz
Lorentz_transformation
All curves of constant width have the same perimeter
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 is a disk
Barbier's_theorem
Length in a vector space
from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance
Norm_(mathematics)
Compact astronomical body
{\displaystyle Q} and the total angular momentum J {\displaystyle J} satisfy the inequality Q 2 4 π ϵ 0 + c 2 J 2 G M 2 ≤ G M 2 {\displaystyle {\frac {Q^{2}}{4\pi
Black_hole
Construct all metric spaces where lines resemble those on a sphere
1930. In 1890, for solving problems on the theory of numbers, Hermann Minkowski introduced a notion of the space that nowadays is called the finite-dimensional
Hilbert's_fourth_problem
Austrian mathematician
von Prechtl Medal (1989) Erwin Schrödinger Prize Minkowski–Hlawka theorem Koksma–Hlawka inequality 10763 Hlawka, an asteroid named after Edmund Hlawka
Edmund_Hlawka
Branch of geometry
Convex bodies: the Brunn-Minkowski theory (2nd ed.). Cambridge: Cambridge University Press. Thompson, A. C. (1996). Minkowski geometry. Cambridge: Cambridge
Convex_geometry
Invariant measure of fractal dimension
is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension. The intuitive concept of dimension of a geometric
Hausdorff_dimension
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
Key result in general relativity
can define the energy-momentum of each infinite region as an element of Minkowski space. Provided that the initial data set is geodesically complete and
Positive_energy_theorem
French mathematician, physicist and engineer (1854–1912)
geometry would entail too much effort for limited profit. So it was Hermann Minkowski who worked out the consequences of this notion in 1907. Like others before
Henri_Poincaré
Hypothetical topological feature of spacetime
taken from Matt Visser's Lorentzian Wormholes (1996).[page needed] If a Minkowski spacetime contains a compact region Ω {\displaystyle \Omega } , and if
Wormhole
Hypothetical FTL transportation by warping space
line of work embeds the bubble in a curved background rather than in Minkowski space. Remo Garattini and Kirill Zatrimaylov found that the gravitational
Alcubierre_drive
English mathematician, mathematical physicist (born 1931)
1967, Penrose developed twistor theory, which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2
Roger_Penrose
Thought experiment in special relativity
constant velocity motion, all of which was visualized by Thirring using Minkowski diagrams. The same result was already before obtained by Einstein (1918)
Twin_paradox
On the existence of hyperplanes separating disjoint convex sets
axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological
Hyperplane_separation_theorem
Swiss mathematician (1908–1981)
Weitzenböck's inequality and was generalized in turn by Pedoe's inequality. In the same 1937 paper in which Hadwiger and Finsler published this inequality, they
Hugo_Hadwiger
Type of mathematical plane curve
Aleksandrov-Fenchel inequality under regularity assumptions, Monatshefte für Mathematik 182 (2017), 65-76 R. Schneider.: Convex Bodies: The Brunn–Minkowski Theory
Hedgehog_(geometry)
Concept in geometry
perimeter that encloses the maximum area. This is known as the isoperimetric inequality, which states that if a rectifiable Jordan curve in the Euclidean plane
Area_of_a_circle
Concept in number theory
Another simple proof of the Dirichlet's approximation theorem is based on Minkowski's theorem applied to the set S = { ( x , y ) ∈ R 2 : − N − 1 2 ≤ x ≤ N
Dirichlet's approximation theorem
Dirichlet's_approximation_theorem
Type of function in linear algebra
the origin in a topological vector space X {\displaystyle X} then the Minkowski functional of U , {\displaystyle U,} p U : X → [ 0 , ∞ ) , {\displaystyle
Sublinear_function
Sides and diagonals have equal sums of squares
(1734–1798) Inner product space – Vector space with generalized dot product Minkowski distance – Vector distance function Normed vector space – Vector space
Parallelogram_law
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