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...

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...

f+g} is in L p ( S ) , {\displaystyle L^{p}(S),} and we have the triangle inequality ‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p {\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}...

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

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 )...

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner...

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

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

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...

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...

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

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...

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

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...

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...

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...

{\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)}...

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

Golden triangle (mathematics) Gossard perspector Hadwiger–Finsler inequality Heilbronn triangle problem Heptagonal triangle Heronian triangle Heron's...

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

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

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}...

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;...

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...

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...

classical triangle inequalities", Forum Geometricorum, 12: 197–209; see p. 198 Pambuccian, Victor; Schacht, Celia (2018), "Euler's inequality in absolute...

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 Ruzsa triangle inequality is an important tool which is used to generalize Plünnecke's inequality to the Plünnecke–Ruzsa inequality. Its statement...