we obtain a sucint and nesessery theoreoms simple for compactness andweakly compactness of the best approximate sets by closed unit balls. also weconsider relations kadec-klee property and shur property with this objects.these theorems are extend of papers mohebi and narayana.

We obtain a sucint and nesessery theoreoms simple for compactness andweakly compactness of the best approximate sets by closed unit balls. Also weconsider relations Kadec-Klee property and shur property with this objects.These theorems are extend of papers mohebi and Narayana.

Let D be a set of n pairwise disjoint unit balls in R and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at least m points from P . This generalizes the notion of halving hyperplanes, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for ins...

In section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4...

We investigate the familyM of intersections of balls in a finite dimensional vector space with a polyhedral norm. The spaces for whichM is closed under Minkowski addition are completely determined. We characterize also the polyhedral norms for which M is closed under adding a ball. A subset P ofM consists of the Mazur sets K, defined by the property that for any hyperplane H not meeting K there...

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak * denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed po...

Given two open unit balls B1 and B2 in complex Banach spaces, we consider a holomorphic mapping f : B1 → B2 such that f(0) = 0 and f ′(0) is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that f(B1) is a complex closed analytic submanifold of B2.

We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...

We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...

1. INTRODUCTION If F is a finite family of sets, then the intersection graph r(F) is the graph with vertex-set F and edges the unordered pairs C, D of distinct elements of F such that C n D # 0. It is easy to see [6, p. 19] that every graph G is isomorphic to some intersection graph T(F). Some interesting classes of graphs have arisen by letting F range over families of balls in some metric spa...