We show the power of posets in computational geometry by solving several problems posed on a set S of n points in the plane: (1) nd the n ? k ? 1 rectilinear farthest neighbors (or, equivalently, k nearest neighbors) to every point of S (extendable to higher dimensions), (2) enumerate the k largest (smallest) rectilinear distances in decreasing (increasing) order among the points of S, (3) give...

In this paper we investigate the property (HLUR), a generalisation of (LUR) Banach space. A space having (HLUR) is called an HLUR We characterise with help known geometric properties and study various spaces. show that for any finite dimensional space, coincides anti-Daugavet also some applications spaces in connection farthest points sets.

(a) If one wants to de ne the locus where p is further away than all other points, i.e., strictly the farthest point, consider the following: As h(p, p ) := {x 2 R | d(x, p) < d(x, p )} denotes the open halfplane of all points that are strictly closer to p than to p , h(p , p) is the open halfplane on the other side of the bisector b(p, p ) of p and p that contains all points strictly farther a...

Abstract— The term ‘Clustering’ means grouping of input datasets into subsets, which is commonly called ’clusters’ .The elements in the clusters, are somewhat similar. Many clustering algorithms require the specification of the number of clusters to be produced, prior to execution of the algorithm. The specified method proposes a simple and efficient clustering method. This paper presents an ov...

The aim of this paper is to present some existence results for nearest-point and farthestpoint problems, in connection with some geometric properties of Banach spaces. The idea goes back to Efimov and Stečkin who, in a series of papers (see [28, 29, 30, 31]), realized for the first time that some geometric properties of Banach spaces, such as strict convexity, uniform convexity, reflexivity, an...

For the kind of coverings codes called multiple coverings of the farthestoff points (MCF) we define μ-density as a characteristic of quality. A concept of multiple saturating sets ((ρ, μ)-saturating sets) in projective spaces PG(N, q) is introduced. A fundamental relationship of these sets with MCF codes is showed. Lower and upper bounds for the smallest possible cardinality of (1, μ)-saturatin...

In this paper we deal with the special class of covering codes consisting of multiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the μ-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main result...