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Property of geometry, also used to generalize the notion of "distance" in metric spaces
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length
Triangle_inequality
geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain
List_of_triangle_inequalities
In additive combinatorics, the Ruzsa triangle inequality, also known as the Ruzsa difference triangle inequality to differentiate it from some of its
Ruzsa_triangle_inequality
Type of metric space
mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d ( x , z ) ≤ max { d ( x , y ) , d ( y , z )
Ultrametric_space
Similarity measure for number sequences
triangle inequality property — or, more formally, the Schwarz inequality — and it violates the coincidence axiom. To repair the triangle inequality property
Cosine_similarity
Shape with three equal sides
altitude formula. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter
Equilateral_triangle
Relation between distances of four points
the quadrilaterals must obey the triangle inequality. As a special case, Ptolemy's theorem states that the inequality becomes an equality when the four
Ptolemy's_inequality
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
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
Function spaces generalizing finite-dimensional p norm spaces
two vectors is no larger than the sum of lengths of the vectors (triangle inequality). Abstractly speaking, this means that R n {\displaystyle \mathbb
Lp_space
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
Inequality_(mathematics)
Mathematical space with a notion of distance
to x: d ( x , y ) = d ( y , x ) {\displaystyle d(x,y)=d(y,x)} The triangle inequality holds: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {\displaystyle d(x
Metric_space
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
geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain
Barrow's_inequality
Length in a vector space
distance 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)
Shape with three sides
is the matrix determinant. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to
Triangle
Triangles without a right angle
acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle
Acute_and_obtuse_triangles
NP-hard problem in combinatorial optimization
the triangle inequality. A very natural restriction of the TSP is to require that the distances between cities form a metric to satisfy the triangle inequality;
Travelling_salesman_problem
Mathematical statistics distance measure
contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence
Kullback–Leibler_divergence
inequality Hoffman-Wielandt inequality Peetre's inequality Sylvester's rank inequality Triangle inequality Trace inequalities Bendixson's inequality Weyl's
List_of_inequalities
Approximation for the travelling salesman problem
the distances form a metric space (they are symmetric and obey the triangle inequality). It is an approximation algorithm that guarantees that its solutions
Christofides_algorithm
Mode of convergence of an infinite series
if its tails can be made arbitrarily small in norm—and apply the triangle inequality. In particular, for series with values in any Banach space, absolute
Absolute_convergence
Measure of similarity and diversity between sets
is easy to construct an example which disproves the property of triangle inequality. Tanimoto distance is often referred to as a synonym for Jaccard
Jaccard_index
Distance from zero to a number
numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a
Absolute_value
} The Ruzsa triangle inequality is an important tool which is used to generalize Plünnecke's inequality to the Plünnecke–Ruzsa inequality. Its statement
Plünnecke–Ruzsa_inequality
Geometric inequality applicable to any closed curve
triangle. This is implied, via the AM–GM inequality, by a stronger inequality which has also been called the isoperimetric inequality for triangles:
Isoperimetric_inequality
Perpendicular line segment from a triangle's side to opposite vertex
In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite
Altitude_(triangle)
Measure of distance between two clusterings related to mutual information
variation of information is a true metric, in that it obeys the triangle inequality. Suppose we have two partitions X {\displaystyle X} and Y {\displaystyle
Variation_of_information
Mathematical theorem
Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality
Kantorovich_inequality
Triangle area in terms of side lengths
lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real
Heron's_formula
Separation between two points
always the same as the distance from y to x. Distance satisfies the triangle inequality: if x, y, and z are three objects, then d ( x , z ) ≤ d ( x , y )
Distance
Mathematical description of spacetime used in relativity
is always positive. This can be seen from the reversed Cauchy–Schwarz inequality below. It follows that if the scalar product of two vectors is zero, then
Minkowski_spacetime
Optimization problem in computer science
expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space
Nearest_neighbor_search
Limiting case which is different from the rest of the class
degenerate triangle is a triangle that does not satisfy the strict triangle inequality: a + b > c {\displaystyle a+b>c} , instead, it satisfies: a + b =
Degeneracy_(mathematics)
Inequality applying to triangles
the pair of triangles. Pedoe's inequality is a generalization of Weitzenböck's inequality, which is the case in which one of the triangles is equilateral
Pedoe's_inequality
Length of a line segment
while the distance from any point to itself is zero. It obeys the triangle inequality: for every three points p {\displaystyle p} , q {\displaystyle q}
Euclidean_distance
Inequality applying to triangles
In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a {\displaystyle a} , b {\displaystyle
Weitzenböck's_inequality
Mathematical statement which always holds true
preferred, as more accurate. Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that c ≤ a + b
Law_(mathematics)
Statistic used for comparing the similarity of two samples
S=2J/(1+J)} . Since the Sørensen–Dice coefficient does not satisfy the triangle inequality, it can be considered a semimetric version of the Jaccard index.
Dice-Sørensen_coefficient
On sums of distances in triangles
In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides
Erdős–Mordell_inequality
Notion in metric geometry
distances from some new point to the points in X that must satisfy the triangle inequality together with the distances in (X,d). The second requirement states
Tight_span
Area of combinatorics in mathematics
{|A-B|}{\sqrt {|A||B|}}}.} The Ruzsa triangle inequality asserts that the Ruzsa distance obeys the triangle inequality: d ( B , C ) ≤ d ( A , B ) + d ( A
Additive_combinatorics
Term in quantum mechanics
{tr} |{\sqrt {\rho }}{\sqrt {\sigma }}|.} Note that this inequality is the triangle inequality applied to the singular values of the matrix. Indeed, for
Fidelity_of_quantum_states
String distance measure
the mathematical sense of that term because it does not obey the triangle inequality. The Jaro similarity sim j {\displaystyle {\text{sim}}_{j}} of two
Jaro–Winkler_distance
Inequality applicable to triangles
the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths
Hadwiger–Finsler_inequality
Statistical measure of biodiversity difference
erroneously called a distance ("A well-defined distance function obeys the triangle inequality, but there are several justifiable measures of difference between
Bray–Curtis_dissimilarity
Vector space on which a distance is defined
{\displaystyle \lVert \lambda x\rVert =|\lambda |\,\lVert x\rVert } Triangle inequality: for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle
Normed_vector_space
On distance between centers of a triangle
classical triangle inequalities", Forum Geometricorum, 12: 197–209; see p. 198 Pambuccian, Victor; Schacht, Celia (2018), "Euler's inequality in absolute
Euler's_theorem_in_geometry
Metric in computer science
→ ABC. Note that for the optimal string alignment distance, the triangle inequality does not hold: OSA(CA, AC) + OSA(AC, ABC) < OSA(CA, ABC), and so
Damerau–Levenshtein_distance
Vector quantization algorithm minimizing the sum of squared deviations
implementation very inefficient. Some implementations use caching and the triangle inequality in order to create bounds and accelerate Lloyd's algorithm. Finding
K-means_clustering
Theorem about triangles
Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is
Ono's_inequality
On all absolute values of rational numbers
i = 1 + 1 + ⋯ + 1 {\displaystyle c_{i}=1+1+\cdots +1} so by the triangle inequality, | c i | ≤ | 1 | + | 1 | + ⋯ + | 1 | = c i ≤ b − 1 {\displaystyle
Ostrowski's_theorem
Triangle with vertices at midpoints of another triangle's sides
geometry, the medial triangle or midpoint triangle of a triangle △ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC, and
Medial_triangle
Generalization of metric spaces in mathematics
, y ) = d ( y , x ) {\displaystyle d(x,y)=d(y,x)} Subadditivity/Triangle inequality: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {\displaystyle d(x,z)\leq
Pseudometric_space
Measure of difference between two points
Bregman divergences are similar to metrics, but satisfy neither the triangle inequality (ever) nor symmetry (in general). However, they satisfy a generalization
Bregman_divergence
Result in measure theory
d\mu } . The proof is based fundamentally on an application of the triangle inequality and Fatou's lemma. Applied to probability theory, Scheffe's theorem
Scheffé's_lemma
Norm on a vector space of matrices
{\displaystyle \|A+B\|\leq \|A\|+\|B\|\ } (sub-additive or satisfying the triangle inequality) The only feature distinguishing matrices from rearranged vectors
Matrix_norm
Relation between sides of a right triangle
to the triangle inequality). The following statements apply: If a2 + b2 = c2, then the triangle is right. If a2 + b2 > c2, then the triangle is acute
Pythagorean_theorem
Combinatorial optimization problem
in a metric space, providing a complete graph that satisfies the triangle inequality. It has application in facility location and clustering. The problem
Metric_k-center
Tree data structure
and B-trees. It is constructed using a metric and relies on the triangle inequality for efficient range and k-nearest neighbor (k-NN) queries. While
M-tree
Topics referred to by the same term
Exact triangle, a collection of objects in category theory Triangle inequality, Euclid's proposition that the sum of any two sides of a triangle is longer
Triangle_(disambiguation)
Real function with secant line between points above the graph itself
= | x | {\displaystyle f(x)=|x|} is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point x = 0. {\displaystyle
Convex_function
Hash based data structure
from B to A are the same it follows the triangle inequality: given A, B and C are vertices (points) of a triangle, then the distance from A to B is shorter
Kademlia
Similarity of two probability distributions
metric, despite being named a "distance", since it does not obey the triangle inequality. Both the Bhattacharyya distance and the Bhattacharyya coefficient
Bhattacharyya_distance
Type of entropy in quantum theory
left side of the triangle inequality above, one can show that the strong subadditivity inequality is equivalent to the following inequality: S ( ρ A ) + S
Von_Neumann_entropy
Absolute value of (x - y), a metric
| x − y | + | y − z | {\displaystyle |x-z|\leq |x-y|+|y-z|} (the triangle inequality); equality holds if and only if x ≤ y ≤ z {\displaystyle x\leq y\leq
Absolute_difference
Overview of and topical guide to geometry
of triangle inequalities List of triangle topics Pedal triangle Pedoe's inequality Pythagorean theorem Pythagorean triangle Right triangle Triangle inequality
Outline_of_geometry
Triangle with at least two sides congruent
In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified
Isosceles_triangle
Diagram used to represent quantum field theory calculations
spin numbers a, b, and c. Then, these requirements are stated as: Triangle inequality: a ≤ b + c and b ≤ a + c and c ≤ a + b. Fermion conservation: a +
Spin_network
Tree data structure for metric spaces
so far by taking advantage of the BK-tree organization and of the triangle inequality (cut-off criterion). Input: t {\displaystyle t} : the BK-tree; d
BK-tree
otherwise estimating the modulus of the exponential sum by means of the triangle inequality as ≤ N {\displaystyle \scriptstyle \leq \,N} provides a better bound
Weyl's inequality (number theory)
Weyl's_inequality_(number_theory)
Prediction of digital video quality
identity of indiscernibles, and symmetry properties, but not the triangle inequality or non-negativity, and thus is not a distance function. However,
Structural similarity index measure
Structural_similarity_index_measure
Computer science metric for string similarity
than the sum of their Levenshtein distances from a third string (triangle inequality). An example where the Levenshtein distance between two strings of
Levenshtein_distance
Type of function in linear algebra
to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by ‖ x + y ‖ ≤ K ( ‖ x ‖ + ‖ y ‖ ) {\displaystyle \|x+y\|\leq
Quasinorm
Type of vector space in math
must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths
Hilbert_space
Neural network working on two input vectors
x , y ) = δ ( y , x ) {\displaystyle \delta (x,y)=\delta (y,x)} Triangle inequality: δ ( x , z ) ≤ δ ( x , y ) + δ ( y , z ) {\displaystyle \delta (x
Siamese_neural_network
Testable implication of local hidden-variable theories
b\right)\right|} (by the triangle inequality again), which is the CHSH inequality. In their 1974 paper, Clauser and Horne show that the CHSH inequality can be derived
CHSH_inequality
Fundamental space of geometry
metric, as it is positive definite, symmetric, and satisfies the triangle inequality d ( P , Q ) ≤ d ( P , R ) + d ( R , Q ) . {\displaystyle d(P,Q)\leq
Euclidean_space
Algorithm for measuring similarity between temporal sequences
distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences
Dynamic_time_warping
Mathematical function
seminorm if it satisfies the following two conditions: Subadditivity/Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)}
Seminorm
Formulas about vectors in three-dimensional Euclidean space
The triangle inequality: ‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|\mathbf {A+B} \|\leq \|\mathbf {A} \|+\|\mathbf {B} \|} The reverse triangle inequality:
Vector_algebra_relations
Construct all metric spaces where lines resemble those on a sphere
of congruence that involve the concept of the angle dropped, and 'triangle inequality', regarded as an axiom, added. If one assumes the continuity axiom
Hilbert's_fourth_problem
Circle that passes through the vertices of a triangle
circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius
Circumcircle
Least-weight tree connecting graph vertices
requirement for edge lengths to obey normal rules of geometry such as the triangle inequality. A spanning tree for that graph would be a subset of those paths
Minimum_spanning_tree
Distance metric used for comparing biological communities
only when entities are identical, transitive, and conforming to the triangle inequality. If there are several different samples, a distance matrix can be
UniFrac
Tree data structure
spaces. Metric trees exploit properties of metric spaces such as the triangle inequality to make accesses to the data more efficient. Examples include the
Metric_tree
Type of metric space in mathematics
( x , y ) , {\displaystyle d(x,z)+d(z,y)=d(x,y),\,} that is, the triangle inequality becomes an equality. A metric space is called convex (or Menger-convex
Convex_metric_space
Metric that measures the distance between two strings of text
metric (e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to
String_metric
Triangle whose side lengths and area are integers
Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. Heronian triangles are named
Heronian_triangle
Function in algebra
group homomorphism on K×. The third property is a version of the triangle inequality on metric spaces adapted to an arbitrary Γ (see § Multiplicative
Valuation_(algebra)
characterizing automedian triangles, but they would not satisfy the triangle inequality and could not be used to form the sides of a triangle. Consequently, using
Automedian_triangle
Vector distance function
formula does not define a metric because it fails to satisfy the triangle inequality. For example, distance between ( 0 , 0 ) {\displaystyle (0,0)} and
Minkowski_distance
Form of mathematical proof
{\displaystyle P(k)} is true. Using the angle addition formula and the triangle inequality, we deduce: | sin ( k + 1 ) x | = | sin k x cos x + sin x
Mathematical_induction
Mathematical property of algebraic structures
satisfies the stronger condition, referred to as the ultrametric triangle inequality, | x + y | ≤ max ( | x | , | y | ) , {\displaystyle |x+y|\leq \max(|x|
Archimedean_property
Mathematical theorem
was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration). The original
Paley–Wiener_theorem
Average value of a random variable
the formula |X| = X+ + X− as discussed above, together with the triangle inequality, it follows that for any random variable X {\displaystyle X} with
Expected_value
Theorem relating the diameter of a point set to the minimum radius of an enclosing ball
The first inequality is implied by the triangle inequality for the center of the ball and the two diametral points, and the second inequality follows since
Jung's_theorem
Type of function in linear algebra
~p(-x),} so if p {\displaystyle p} is also symmetric then the reverse triangle inequality will hold for all vectors x , y ∈ X , {\displaystyle x,y\in X,} |
Sublinear_function
Point in a triangle that can be seen as its middle under some criteria
Furthermore, in order to represent the sides of a triangle they must satisfy the triangle inequality. So, in practice, every function's domain is restricted
Triangle_center
Distance between two metric-space subsets
compact, then d(X, Y) will be finite; d(X, X) = 0; and d inherits the triangle inequality property from the distance function in M. As it stands, d(X, Y) is
Hausdorff_distance
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