On the metric triangle inequality

The Poverty-Growth-Inequality Triangle

The Triangle Inequality and Its Applications in the Relative Metric Space

Let C be a plane convex body. For arbitrary points , denote by , n a b E  ab the Euclidean length of the line-segment . Let be a longest chord of C parallel to the line-segment . The relative distance between the points and is the ratio of the Euclidean distance between and b to the half of the Euclidean distance between and . In this note we prove the triangle inequality in with the relative ...

Triangle inequality for resistances

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function y∗ e = y r e/μ s e. In this formula, ye is the potential difference and y∗ e current in e, while μe is the resistance of e, while r and s are two strictly positive real parameters common for all edges. In 1987, Gvishiani and Gurvich [4] proved that, for every two nodes a, b of the ci...

A note on the triangle inequality for the Jaccard distance

Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed. The Jaccard index [8] is a classical similarity measure on sets with a lot of practical applications in information retrieval, data mining, machine learning, and many more (cf., e.g., [7]). Measuring the relative size of the overlap of two finite...

ON INEQUALITY Rp < R OF THE PEDAL TRIANGLE

In this paper we give a simple proof of the pedal triangle inequality Rp < R , where R is the circumradius of a triangle and Rp is the circumradius of the pedal triangle of an interior point with respect to this triangle. We also establish a stronger result and a refinement of inequality Rp < R . Some related interesting conjectures checked by the computer are put forward. Mathematics subject c...

Concept Dissimilarity with Triangle Inequality

Several researchers have developed properties that ensure compatibility of a concept similarity or dissimilarity measure with the formal semantics of Description Logics. While these authors have highlighted the relevance of the triangle inequality, none of their proposed dissimilarity measures satisfy it. In this work we present a theoretical framework for dissimilarity measures with this prope...