Knowra Triangle inequality Triangle inequality The triangle inequality states that the distance between two points is no greater than the sum of their distances to a third point. It also constrains lengths in metric spaces and norms.
Metric space : A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. The inequality is one of the axioms that makes a distance function a metric.
Euclidean distance : The straight-line distance between points in Euclidean space, computed from coordinate differences by the Pythagorean theorem. The familiar geometric version compares one straight segment with two sides of a triangle.
Lipschitz continuity : A condition that bounds changes in a function's output by a constant multiple of changes in its input. Triangle-inequality estimates often prove that a function cannot change too quickly.
Squared Euclidean distance : The sum of squared coordinate differences between two points, without taking the square root. It can violate the triangle inequality, so it is not itself a metric.
Normed vector space : A vector space equipped with a norm that measures vector length and satisfies homogeneity and the triangle inequality. The norm form, ||x + y|| ≤ ||x|| + ||y||, extends the rule to vectors.
Absolute value : A real number's nonnegative magnitude, equal to its distance from zero on the number line. On the real line, the triangle inequality becomes |x + y| ≤ |x| + |y|.
Convergence in probability : A mode of convergence in which the probability of a fixed-size deviation tends to zero. Probability bounds split a large deviation into smaller events using the triangle inequality.
Hamming distance : The number of positions at which two finite strings of equal length differ. Unlike squared distance, it satisfies the triangle inequality and defines a metric.
Cauchy–Schwarz inequality : An inequality bounding the absolute inner product of two vectors by the product of their lengths. Together with inner-product identities, it proves the triangle inequality for Euclidean distance.
Vector norm : A function assigning nonnegative length to vectors while satisfying definiteness, homogeneity, and the triangle inequality. The rule is a defining requirement for vector length generalized beyond ordinary geometry.
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