Abstract:
In many applications such as web-based search, document summarization, facility location and other applications, the results are preferable to be both representative and diversified subsets of documents. The goal of this study is to select a good "quality", bounded-size subset of a given set of items, while maintaining their diversity relative to a semi-metric distance function. This problem was first studied by Borodin et al\cite{borodin}, but a crucial property used throughout their proof is the triangle inequality. In this modified proof, we want to relax the triangle inequality and relate the approximation ratio of max-sum diversification problem to the parameter of the relaxed triangle inequality in the normal form of the problem (i.e., a uniform matroid) and also in an arbitrary matroid.

Abstract:
This is a short note related to the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev. It contains a general "geometric Ruzsa triangle inequality" which is then applied to get the same kind of inequality in a metric space with dilations.

Abstract:
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.

Abstract:
Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.

Abstract:
We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.

Abstract:
We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak--strong uniqueness implies weak--strong uniqueness in law.

Abstract:
Some reverses for the generalised triangle inequality in complex inner product spaces that improve the classical Diaz-Metcalf results and applications are given.

Abstract:
In this paper we will demonstrate a new inequality between sine and a function of first degree. Applications of this inequality in triangle are given.

Abstract:
In this paper, we give two new simpler proofs of a sharp inequality for the medians of a triangle. We also establish two new inequalities by using this sharp inequality. Some related conjectures checked by the computer are put forward, which include two conjectures related to the famous Erd s-Mordell inequality.