A geometric approach of Blundon’s inequality is presented. Theorem 2.1 gives the formula for cos ̂ ION in terms of the symmetric invariants s , R , r of a triangle, implying Blundon’s inequality (Theorem 2.2). A dual formula for cos ̂ IaONa is given in Theorem 3.1 and this implies the dual Blundon’s inequality (Theorem 3.2). As applications, some inequalities involving the exradii of the triangle...

We give deterministic versions of randomized approximation algorithms for several ranking and clustering problems that were proposed by Ailon, Charikar and Newman[1]. We show that under a reasonable extension of the triangle inequality in clustering problems, we can resolve Ailon et al.’s open question whether there is an approximation algorithm for weighted correlation clustering with weights ...

Bailey and Bannister [College Math. Journal, 28 (1997) 182–186] proved that a stronger triangle inequality holds in the Euclidean plane for all triangles having largest angle less than arctan( 7 ) ≈ 74◦. We use hyperbolic trigonometry to show that a stronger triangle inequality holds in the hyperbolic plane for all triangles having largest angle less than or equal to 65.87◦ .

We discuss the extension of inequality RA c a rb + b a rc to the plane of triangle ABC . Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdös-Mordell inequality, and some inequalities of Erdös-Mordell type. Mathematics subject classification (2010): 51M16, 51M04, 14H50.

Some new reverses for the generalised triangle inequality in inner product spaces and applications are given. Applications in connection to the Schwarz inequality are provided as well.

Similarity search is a fundamental problem for many data analysis techniques. Many efficient techniques rely on the triangle inequality of metrics, which allows pruning parts space based transitive bounds distances. Recently, cosine similarity has become popular alternative choice to standard Euclidean metric, in particular context textual and neural network embeddings. Unfortunately, not metri...