In this paper generalized triangle inequality and its reverse in a p-Fréchet space where, 0 < p < 1 are obtained.

In the problem of the month 1999 : 106], one was to prove that p a + b ; c + p b + c ; a + p c + a ; b p a + p b + p c , where a, b, c are sides of a triangle. It is to be noted that this inequality will follow immediately from the Majorization Inequality 1]. Here, if A and B are vectors (a 1 a 2 : : : a n), we say that A majorizes B and write it as A B. Then, if F is a convex function, If F is...

Validity of the triangle inequality and minimality, both axioms for twowaydissimilarities, ensures that a two-waydissimilarity is nonnegative and symmetric. Three-way generalizations of the triangle inequality andminimality from the literature are reviewed and it is investigated what forms of symmetry and nonnegativity are implied by the three-way axioms. A special form of three-way symmetry th...

While Bregman divergences [3] have been used for several machine learning problems in recent years, the facts that they are asymmetric and does not satisfy triangle inequality have been a major limitation. In this paper, we investigate the relationship between two families of symmetrized Bregman divergences and metrics, which satisfy the triangle inequality. Further, we investigate kmeans-type ...

Given an undirected graph with edge costs and edge demands, the Capacitated Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity vehicles so as to satisfy all demands. Constant-factor polynomial-time approximation algorithms were proposed for CARP with triangle inequality, while CARP was claimed to be NP-hard to approximate within any constant factor in general. Correcting...

was first discovered by M. Petrovich in 1917, [5] (see [4, p. 492]) and subsequently was rediscovered by other authors, including J. Karamata [2, p. 300 – 301], H.S. Wilf [6], and in an equivalent form by M. Marden [3]. The first to consider the problem of obtaining reverses for the triangle inequality in the more general case of Hilbert and Banach spaces were J.B. Diaz and F.T. Metcalf [1] who...

The k-center problem with triangle inequality is that of placing k center nodes in a weighted undirected graph in which the edge weights obey the triangle inequality, so that the maximum distance of any node to its nearest center is minimized. In this paper, we consider a generalization of this problem where, given a number p, we wish to place k centers so as to minimize the maximum distance of...

A new solution method to the Nearest Neighbour Problem is presented. The method is based upon the triangle inequality and works well for small point sets, where traditional solutions are particularly ineffective. Its performance is characterized experimentally and compared with k-d tree and Elias approaches. A hybrid approach is proposed wherein the triangle inequality method is applied to the ...

Given the complete graph Kn = (V,E) on n nodes with edge costs c ∈ R, the symmetric traveling salesman problem (henceforth TSP) is to find a Hamilton cycle (or tour) in Kn of minimum cost. This problem is known to be NP-hard, even in the case where the costs satisfy the triangle inequality, i.e. when cij + cjk ≥ cik for all i, j, k ∈ V (see [5]). When the costs satisfy the triangle inequality, ...

In this paper we present some new inequalities in normed linear spaces which much improve the triangle inequality. Our results refine and generalize the corresponding ones obtained by Mitani et al. [ On sharp triangle inequalities in Banach spaces, J. Math. Anal. Appl. 336 (2007) 1178-1186]. Mathematics subject classification (2010): Primary 47A30; Secondary 26D20.