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The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces (p ≥ 1), and inner product spaces.
Mathematical inequality relating inner products and norms. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely ...
Cosine similarity. In data analysis, cosine similarity is a measure of similarity between two non-zero vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not ...
The Euclidean distance obeys the triangle inequality, so the Euclidean TSP forms a special case of metric TSP. However, even when the input points have integer coordinates, their distances generally take the form of square roots , and the length of a tour is a sum of radicals , making it difficult to perform the symbolic computation needed to ...
Euclidean distance. In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek ...
For a fixed length n, the Hamming distance is a metric on the set of the words of length n (also known as a Hamming space), as it fulfills the conditions of non-negativity, symmetry, the Hamming distance of two words is 0 if and only if the two words are identical, and it satisfies the triangle inequality as well: [2] Indeed, if we fix three words a, b and c, then whenever there is a ...
Vector algebra relations. The following are important identities in vector algebra. Identities that only involve the magnitude of a vector and the dot product (scalar product) of two vectors A · B, apply to vectors in any dimension, while identities that use the cross product (vector product) A × B only apply in three dimensions, since the ...
The simplest counterexample of this is given by the three sets {a}, {b}, and {a,b}, the distance between the first two being 1, and the difference between the third and each of the others being one-third. To satisfy the triangle inequality, the sum of any two of these three sides must be greater than or equal to the remaining side. However, the ...