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...

geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain...

In additive combinatorics, the Ruzsa triangle inequality, also known as the Ruzsa difference triangle inequality to differentiate it from some of its...

triangle inequality property — or, more formally, the Schwarz inequality — and it violates the coincidence axiom. To repair the triangle inequality property...

equilateral triangle are 60°, the formula is as desired.[citation needed] A version of the isoperimetric inequality for triangles states that the triangle of greatest...

Minkowski inequality establishes that the Lp spaces satisfy the triangle inequality in the definition of normed vector spaces. The inequality is named...

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 )...

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...

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...

product fg is in L1(μ). Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), and also to...

triangle. This is implied, via the AM–GM inequality, by a stronger inequality which has also been called the isoperimetric inequality for triangles:...

two vectors is no larger than the sum of lengths of the vectors (triangle inequality). Abstractly speaking, this means that R n {\displaystyle \mathbb...

the distances form a metric space (they are symmetric and obey the triangle inequality). It is an approximation algorithm that guarantees that its solutions...

Kantorovich inequality Kunita–Watanabe inequality Minkowski inequality – Triangle inequality in Lp Paley–Zygmund inequality – Probability equation in mathematics...

{\displaystyle r>0} and all x ∈ X . {\displaystyle x\in X.} Subadditivity/Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)}...

the quadrilaterals must obey the triangle inequality. As a special case, Ptolemy's theorem states that the inequality becomes an equality when the four...

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...

inequality Hoffman-Wielandt inequality Peetre's inequality Sylvester's rank inequality Triangle inequality Trace inequalities Bendixson's inequality Weyl's...

distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance...

Markov's inequality Minkowski inequality Nesbitt's inequality Pedoe's inequality Poincaré inequality Samuelson's inequality Sobolev inequality Triangle inequality...

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...

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;...

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...

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...

geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain...

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...

Kantorovich inequality is a particular case of the Cauchy–Schwarz inequality, which is itself a generalization of the triangle inequality. The triangle inequality...

numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a...

contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence...

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...