We investigate the enumerative geometry aspects of algorithmic line problems when the admissible bodies are balls or polytopes. For this purpose, we study the common tangent lines/transversals to k balls of arbitrary radii and 4− k lines in R3. In particular, we compute tight upper bounds for the maximum number of real common tangents/transversals in these cases. Our results extend the results ...

We show that in the nite dimensional space R provided with a metric induced by a norm the collection of Borel sets is the smallest collection containing the open balls and closed under complements and countable disjoint unions

In this article we give general conditions on ametric space to insure that the poset of closed formal balls is an FS-domain. © 2008 Elsevier B.V. All rights reserved.

We deal with the isoperimetric and the shift problem for subsets of measure one half in product probability spaces. We prove that the canonical central half-spaces are extremal in particular cases: products of log-concave measures on the real line satisfying precise conditions and products of uniform measures on spheres, or balls. As a corollary, we improve the known log-Sobolev constants for E...

We obtain some results in both, Lorentz and Finsler geometries, by using a correspondence between the conformal structure of standard stationary spacetimes on M = R × S and Randers metrics on S. In particular: (1) For stationary spacetimes: we give a simple characterization on when R×S is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface),...

In this paper, we establish some common fixed point theorems for fuzzy mappings satisfying fuzzy contractive conditions which are general than Chatterjee type and Kannan type fuzzy contractive conditions on closed balls in a complete metric space. Our results generalize the corresponding results in the current literature. AMS (2000) subject classification: 45S40 • 47H10 • 54H25

The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices whic...

Abstraca. The main object of the paper is to study the distance betwecn Banach spaces introduced by Kadets. For Banach spaccs Xand y. thc lGders distancc is denned to be rhe infimum of the Hausdorfl distance d(Bx, rr) betwecn the respoctive closed unit balls over all isomctric linear embeddings of f and yinto a common Banach space Z. This is comparcd with the Gromov-Hausdorff distance which is ...

Conversely, for every -packing subset t1, . . . , tn of T , the closed balls B(ti, /2), i = 1, . . . , n are disjoing and hence every /2-cover of T must have one point in each of the balls B(ti, /2). As a result, an /2-cover of T must have at least n points. This implies that M( /2, T ) ≥ N( , T ). Lemma 1.2 (Volumetric Argument). Let T = X denote the ball in R of radius Γ centered at the origi...