Lower bounds to helly numbers of line transversals to disjoint congruent balls

A line ℓ is a transversal to a family F of convex objects in ℝ ^d if it intersects every member of F. In this paper we show that for every integer d ⩾ 3 there exists a family of 2d−1 pairwise disjoint unit balls in ℝ ^d with the property that every subfamily of size 2d − 2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957. Crucial to the proof is the notion of a pinned transversal, which means an isolated point in the space of transversals. Here we investigate minimal pinning configurations and construct a family F of 2d−1 disjoint unit balls in ℝ ^d with the following properties: (i) The space of transversals to F is a single point and (ii) the space of transversals to any proper subfamily of F is a connected set with non-empty interior.     

Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.     

Centers of symmetry in finite intersections of balls in Banach spaces

It is proved that a real Banach spaceX is aG-space (C _σ - space)if and only if the non-empty intersection of three balls with equal radii (any three balls) has a center of symmetry.     

Convex sets which are intersections of closed balls

In this paper we aim to investigate different questions concerning the stability of the set of all intersections of closed balls in a normed space. We are mainly concerned with: (i) the stability of under the closure of the vector sums; (ii) the stability under the addition of balls. We prove that (i) and (ii) are different properties which have strong connections with the geometry of the space. They have interest both in finite and infinite dimension. In the former case, there is a link with linear programming theory. We also study two more stability properties related to the well-known binary intersection property. Mazur sets and Mazur spaces are introduced, as a natural family satisfying (i). We prove that every two-dimensional normed space is a Mazur space, a result which distinguishes dimension d?2 from dimension d?3. We also discuss the connections between Mazur spaces and porosity.     

Random embeddings of Euclidean spaces in sequence spaces

It is proved that for some absolute constantd and forn≦dm mostn×m matrices with ± 1 entries are good embeddings ofl ₂ ⁿ intol ₁ ^m . Similar theorems are obtained wherel ₁ ^m is replaced by members of a wide class of sequence spaces. Supported in part by NSF Grant No. MCS-79-03042.     

Embedding theorems for Bergman spaces in quasiconformal balls

The authors prove some embedding theorems for Bergman type spaces of functions defined on quasiconformal balls inR ⁿ,n≥2. This article was processed by the author using the Springer-Verlag T_EX mamath marco package 1990     

OnG-semidifferentiable functions in Euclidean spaces

We give a criterion for a functionf:R ⁿ R to be upperG-semidifferentiable in the sense of Ref. 1 at a point . Using this result, we describe upperG-semiderivatives whenG is, for instance, one of the following basic classes of homogeneous functions: the set of all continuous positively homogeneous functions, the set of differences of two sublinear functions, and the set of sublinear functions. As a result, connections between upperG-semidifferentiability and the concepts of differentiability in Refs. 2–4 are obtained.This research was supported by a grant from the World Laboratory. The author would like to thank Professor M. Pappalardo for useful comments.     

g-functions andg-derivatives in Euclidean spaces

In this paper we give several results showing the correspondence between g-functions and g-derivatives with the usual concepts in Euclidean spaces. Entrata in Redazione il 16 novembre 1970.     

On quasiplanes in Euclidean spaces

A variational inequality for the images of -dimensional hyperplanes under quasiconformal maps of the -dimensional Euclidean space is proved when

     

How to collect balls moving in the Euclidean plane

In this paper, we study how to collect n balls moving with a fixed constant velocity in the Euclidean plane by k robots moving on straight track-lines through the origin. Since all the balls might not be caught by robots, differently from Moving-target TSP, we consider the following 3 problems in various situations: (i) deciding if k robots can collect all n balls; (ii) maximizing the number of the balls collected by k robots; (iii) minimizing the number of the robots to collect all n balls. The situations considered in this paper contain the cases in which track-lines are given (or not), and track-lines are identical (or not). For all problems and situations, we provide polynomial time algorithms or proofs of intractability, which clarify the tractability-intractability frontier in the ball collecting problems in the Euclidean plane.     

Uniqueness questions for two-dimensional surfaces in Euclidean spaces

A geometric approach to the solution of uniqueness problems for two-dimensional surfaces with nonzero Gaussian curvature in E⁴ is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 243–256, 1977.The author thanks Professor É. G. Poznyak profoundly for help.     

Upper bounds for Euclidean minima of algebraic number fields

The aim of this paper is to give new upper bounds for Euclidean minima of algebraic number fields. In particular, to show that Minkowski's conjecture holds for the maximal totally real subfields of cyclotomic fields of prime power conductor.     

Shadows and intersections in vector spaces

We prove a vector space analog of a version of the Kruskal-Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erd?s-Ko-Rado theorem for vector spaces.     

Rectification of sets in Euclidean spaces

We criticize traditional definitions of the arc length which require semi-continuity from below. Symmetric definitions of lower and uppern-lengths (n-dimensional volumes) are introduced for a wide class of sets in Euclidean spaces, and the additivity of both functionals is proved.     

Equi-isoclinic planes of Euclidean spaces

In a Euclidean space, a p-set of equi-isoclinic planes is a set of p isoclinic planes of which each pair has the same non-zero angle.The Euclidean 4-space E⁴ contains a unique congruence class of quadruples of equi-isoclinic planes, whereas quintuples of equi-isoclinic planes do not exist in E⁴.In the following a method is given to derive sets of equi-isoclinic planes in Euclidean spaces. We find again the well-known sets of equi-isoclinic planes of E⁴. The quadruples of equi-isoclinic planes in E⁵ are derived. It turns out that E⁵ contains one congruence class of such quadruples which are not flat quadruples and one congruence class of quintuples of equi-isoclinic planes, whereas sextuples of equi-isoclinic planes do not exist in E⁵.It appears that the symmetry group of that quintuple is isomorphic to the symmetric group S₅.     

Embedding metric spaces in Euclidean space

In this paper, we give necessary and sufficient conditions for embedding a given metric space in Euclidean space. We shall introduce the notions of flatness and dimension for metric spaces and prove that a metric space can be embedded in Euclidean n-space if and only if the metric space is flat and of dimension less than or equal to n.