In this paper we consider a new class of continuous location problems where the ''distances'' are measured by gauges of closed (not necessarily bounded) convex sets. These distance functions do not satisfy the definiteness property and therefore they can be used to model those situations where there exist zero-distance regions. We prove a geometrical characterization of these measures of distan...

Let T : X → Y be a bounded linear map between Banach spaces X and Y . Let T ∗ : Y ∗ → X∗ be its adjoint. Let BX and BY ∗ be the closed unit balls of X and Y ∗ respectively. We obtain apparently new estimates for the covering numbers of the set T ∗ (BY ∗ ). These are expressed in terms of the covering numbers of T (BX), or, more generally, in terms of the covering numbers of a “significant” subs...

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces X and Y , the Kadets distance is defined to be the infimum of the Hausdorff distance d(B X , B Y) between the respective closed unit balls over all isometric linear embeddings of X and Y into a common Banach space Z. This is compared with the Gromov-Hausdorff distance which is def...

In this paper, we consider two new variants of the unit covering problem in color-spanning set model: Given a set of n points in d-dimensional plane colored with m colors, the MinCSBC problem is to select m points of different colors minimizing the minimum number of unit balls needed to cover them. Similarly, the MaxCSBC problem is to choose one point of each color to maximize the minimum numbe...

This paper extends the order-theoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to deene a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those whic...

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating con...

The packing stability in symplectic geometry was first noticed by Biran [Bir97]: the symplectic obstructions to embed several balls into a manifold disappear when their size is small enough. This phenomenon is known to hold for all closed manifolds with rational symplectic class (see [Bir99] for the 4-dimensional case, and [BH11, BH13] for higher dimensions), as well as ellipsoids [BH13]. In th...

This paper extends the domain-theoretic approach to computable analysis to complete metric spaces and Banach spaces. We employ the domain of formal balls to deene a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable seque...

Let P be a set of n points in three dimensions. The 2-center problem for P is to find two closed congruent balls with smallest radius which cover P . We present a randomized algorithm for solving the problem in O ( 1 (1−r∗/r0)3n 2 log n ) expected time, where r∗ is the common radius of the optimal solution balls and r0 is the radius of the smallest enclosing ball of P . This improves the naive ...